Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states

Journal Pre-proofs Research paper Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states Soumaya Elmoussaoui, Wael Chmaisani PII: DOI: Reference:

S0009-2614(20)30124-X https://doi.org/10.1016/j.cplett.2020.137209 CPLETT 137209

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Chemical Physics Letters

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13 December 2019 6 February 2020 7 February 2020

Please cite this article as: S. Elmoussaoui, W. Chmaisani, Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states, Chemical Physics Letters (2020), doi: https://doi.org/10.1016/j.cplett.2020.137209

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Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states Soumaya Elmoussaoui1*, Wael Chmaisani1 1Plateforme

de Recherche et d’Analyse en Sciences de l’Environnement (PRASE), Ecole

Doctorale de Sciences et Technologie (EDST), Université Libanaise, Hadath, Liban. Keywords: Ab-initio calculation, Electronic structure, Multireference configuration interaction, Molecular data, Spin-orbit effect, Dipole moment, Ionic fraction, Einstein coefficients, Rovibrational calculations, Franck-Condon factors. Submitted to: Chemical Physics Letters ------------------------------------------------------------------------------------*Corresponding author: [email protected]

Abstract This work focuses on the electronic structure of the low-lying states of the heavy polar diatomic molecule HgCl. Ab-initio calculations are performed at complete active-space self-consistent field (CASSCF) method followed by multireference configuration interaction (MRCI) method and by invoking Davidson correction (+Q). The potential energy and dipole moment curves are investigated in two stages of calculations; 18 states while neglecting the spin-orbit coupling effect and 36 states considering this effect. Spectroscopic constants are computed for bound states. Vibrational levels, transition dipole moments, Einstein coefficients, radiative lifetimes and other constants are investigated. Comparison with data available in literature demonstrated significant agreements.

1. Introduction Interest in heavy polar molecules has been increasing recently as they are considered potential candidates for future eEDM experiments, i.e., ytterbium monofluoride YbF has been proved to have one of the largest values of Eeff [1]. Theoretically, mercuric monohalide diatomic molecules are similar to their ytterbium counterparts in having an unpaired valence electron outside the closed shell which makes them sensitive to eEDMs in their ground electronic states. However, the former possess values of Eeff that are about five times larger, and this is one of the biggest advantages of mercury monohalides making them good candidates for eEDM experiments. Therefore heavy polar diatomic mercuric halides became qualitatively research appealing as promising test cases for future eEDM examinations [2]. With the impact of the spin-orbit (SO) relativistic effect on molecular properties is known to increase with nuclear charge Z to an extent so that molecules containing heavy elements cannot be described correctly even in a qualitative manner without SO effects especially for excited states. To the best of our knowledge, these effects have not been reported before for high-lying states. From an environmental point of view, many studies have shed light on the importance of ab initio calculations of mercury monohalide radicals in aiding future atmospheric modeling studies by determining the radical’s thermochemistry reactions and spectroscopic parameters [3], [4], [5]. In addition, the theoretical study of Yang et al. [6] on the HgF radical reporting its potential towards laser cooling while referring to our previous electronic structure investigation [7] and the experimental ability to monitor the cooling of molecules having radiative lifetimes as low as 16 ns [8] all together have provided us the impetus to explore the merits of the mercuric monochloride molecule HgCl. Many experimental and theoretical studies have reported about the molecule’s ground state and its B_X transition though few have considered the higher excited states. Wieland [9] was first to measure and identify the band heads of the diatomic HgCl radical which had been later confirmed by both Howell [10] and Cornell [11] upon examining the molecule’s emission band systems for the B2→X2 transition. It was concluded that the band at 36387 cm-1 corresponds to the 0,0 transition of the 2→2 system [10] with 𝜔′′𝑒 = 293.4 cm-1 [11]. The reported spectroscopic data were then analyzed and used to calculate Franck-Condon Factors of the B_X bands [12]. Horne et al. [13] studied the combination reaction of mercury atoms with chlorine atoms from sensitized flash photolysis of Hg-CF3Cl. With kinetic absorption spectroscopy they monitored the band intensities of the 2

2

½←

and 2⅔←2 systems of HgCl. For the B2→X2 system, the strongest transition in

spontaneous and stimulated emission was identified corresponding to 𝜈′ = 0,1,2→𝜈′′ = 22,23,24 respectively [14], [15], [16], [17] and the equilibrium distance of the two states were determined [18]. The radiative lifetime of the B-state =22 ns was evaluated from the Stern-Volmer plots using the least squares method [19] and it was also determined from transition dipole moment calculation in a charge transfer model [20]. For the same system, Franck-Condon factors were computed based on the interpretation of the HgCl laser spectra and from HgCl2 photodissociation [21]. The first ab-initio configuration interaction calculations for the low lying states of HgCl were carried out by Wadt [22] using effective core potentials from which spectroscopic constants of the X and B states were reported. For only the ground state, spectroscopic parameters such as the vibrational frequency, the bond length, the dipole moment of the HgCl free radical were calculated by using the relativistically corrected local density functional method [23], [24], different high-level abinitio methods [24], [25], the normalized elimination of the small component and coupled-cluster theory with triple excitations NESC/CCSD(T) method [26], and by fitting empirical potential functions to the respective RKR curves using correlation coefficients [27]. Gas-phase photolysis study of Hg(I) radical species has been conducted recently to understand the global atmospheric mercury cycle. MRCI+Q calculations were used to plot the potential energy curves of these radical species including spin-orbit effect and the spectroscopic constants of the two lowest bound states are reported [5]. In this work, ab-initio calculations considering 9 valence electrons were carried out to investigate the spin-free potential energy curves (PECs) and the static dipole moment curves (DMCs) of 18 low-lying doublet and quartet S electronic states of HgCl that are correlating with the three lowest dissociation asymptotes. Corresponding energy and dipole moment curves are also probed while considering spin-orbit coupling effect showing 32 states in the Ω = 1/2, 3/2, 5/2, 7/2 representation. For the bound states among all, spectroscopic constants as the transition energy with respect to the ground state minimum Te, the equilibrium internuclear distance Re, the harmonic frequency ωe, the anharmonicity correction ωexe, the rotational constant Be, the permanent dipole moment μe and the percentage ionic factor fionic are calculated. Vibrational levels (with J=0) for those low-lying states in both Λ and Ω representations are also investigated and corresponding vibrational constants Ev, Bv, Dv, Rmin and Rmax are calculated. Transition dipole moments TDM between low-lying doublet states are computed and used to calculate Einstein spontaneous emission coefficients 𝐴21 and radiative lifetimes discrete vibrational levels. We were also interested in calculating Franck-Condon factors between the three lowest states to aid further

researches about the molecule’s potential towards laser cooling. Upon comparing the calculated constants to those available in literature, good accordance is noticed. 2. Computational method For the present work, high level ab-initio calculations of HgCl are carried out by employing the multireference configuration interaction (MRCI) method including single and double excitations. The Davidson correction noted as (+Q) is invoked in order to account for unlinked quadruple clusters, all as implemented in the computational chemistry program MOLPRO [28] while taking the advantage of the graphical user interface GABEDIT [29]. The multiconfiguration reference wavefunctions are generated with the state-averaged complete active-space self-consistent field (CASSCF) method. The mercury atom in the [Xe]4f145d106s2 electronic configuration is treated via the quasi-relativistic effective core potential ECP78MWB [30] as a system of 78 core electrons with the remaining 2 electrons being described by the associated (4s4p1d)/[2s2p1d] valence basis set while ECP10MWB [31] is selected for chlorine atom [Ne]3s23p5 to describe 10 core electrons. Employing relativistic core potentials is one way that MOLPRO takes scalar relativistic effects into account. The molecular valence model Hamiltonian used in the quasi-relativistic ECP that describes Hg accounts for the mass-velocity and Darwin effects through the one component term whereas the corresponding two component includes spin orbit effects [30]. The 7 valence electrons of the Cl atom are treated with the augmented correlation-consistent quadruple-zeta (aug-ccpVQZ) basis set [32] that includes polarized and diffuse functions. Orbital and spin angular momentums are uncoupled to investigate the Λ-S states and coupled for the Ω ones. With the ECP basis set of Cl being designed with a spin-orbit averaged component [31], the ECP spin-orbit operator for the Hg atom is added [30] to consider the spin-orbit coupling (SOC) effect on all the investigated low-lying spin-free states. Among the 97 electrons considered for the mercury monohalide molecule (80 electrons for Hg and 17 for Cl), 88 electrons are described by core functions leaving 9 electrons to be explicitly treated as valence electrons in the molecular calculations. In the C2v symmetry, the active space is selected such that it contains 7 molecular orbitals; 5 (Hg: 6s, 6p0, 7s; Cl: 3s, 3p0) and 2 (Hg: 6p±1; Cl: 3p±1) distributed into irreducible representations a1, b1, b2, and a2 in the following way 5a1, 2b1, 2b2, 0a2 noted by [5, 2, 2, 0]. In the range of the internuclear distance R around the equilibrium position of its ground state, the HgCl molecule is assumed to be mainly ionic with the mercury atom at the origin. The potential energy

curves of the investigated states are constructed by calculating 128 single point energies within the internuclear distance range 1.9Å≤R≤5.6Å. 3. Results and discussion 3.1 Adiabatic potential energy curves By employing the computational method discussed above, eighteen -S states correlating with the three lowest dissociation asymptotes are investigated; 12 doublet states (four 2+, four 2, two 2 and two 2-) and 6 quartet states (two 4+, two 4, one 4 and one 4-). Their potential energy curves (PECs) and dipole moment curves (DMCs) are plotted as function of the internuclear distance and shown in Figures 1, 2. The two lowest states X2+ and (1)2 correlates with the lowest asymptote whose dissociation atomic fragments are Hg (1Sg) + Cl (2Pu). The DMC of the second excited state indicates a polarized behavior at large distances in which the molecule dissociates into ionic fragments Hg+(2S) and Cl-(1S). The second asymptote which lies at 35913 cm-1 above the lowest limit corresponds to a combination of twelve doublet and quartet states; (3)2+, (4)2+, (2)2, (3)2, (1)2, (1)2-, (1)4+, (2)4+, (1)4, (2)4 , (1)4 and (1)4-. At large distances, the molecule being in those states dissociates into fragments in 3Pu and 2Pu atomic states for Hg and Cl, respectively. NIST Atomic Spectra Database [33] reports that the experimental energy difference between the two lowest lying atomic asymptotes for HgCl is 37645 cm-1 with which our calculated value shows a relative difference of 4.60%. The third asymptote, whose atomic fragments are Hg (1Pu) + Cl (2Pu), is correlated with (4)2, (2)2, (2)2- states. This asymptote is calculated to lie at 50942 cm-1 above the lowest limit which corresponds to a relative difference of 5.78% with respect to the experimental value 54069 cm-1 reported by NIST. Among the 18 investigated -S states, eight are found to be bound. The states X2Σ+, (2)2Σ+ and (2)2Π are typical bound states; their PECs show significant depth whereas the remaining bound states show shallower interacting curves. 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, ωexe and Be. Then the permanent dipole moment e and the leading configuration at the equilibrium distance are determined. The calculated constants with the experimental and theoretical values available in the literature are given in Table 1. It also gives the values of the dissociation energy 𝐷00 as estimated in terms of the calculated molecular

spectroscopic constants ωe and ωexe [34]. The calculated ground state dissociation energy is 10165 cm-1 shows significant discrepancy with respect to the values reported in literature in which they range between 7257 cm-1 [5] and 11944 cm-1 [21]. Yet theoretical calculations of reliable dissociation energies 𝐷00 are known to be relatively difficult as discrepancies may arise from different factors related to the limited basis sets, the considered active space, the number of valence electrons, the correlation among core and valence electrons, the interaction among states even upon employing the same theoretical method of calculation. For instance, the difference between the values of dissociation energy estimated using long-range theory in the same manner [18], [35] is 250 cm-1 due to slope differences. However, the enthalpy values provided by the NIST-JANAF thermochemical tables [36] leads a bond dissociation enthalpy (BDE) of 8526±836 cm-1 at 0 K for HgCl [37] around which recent experimental and computational results scatter around. Discussion related to the possible origin of discrepancy in our calculations is provided later in this section. Around the ground state equilibrium, both the equilibrium distance Re=2.331Å and its harmonic frequency ωe =293.6 cm-1 show excellent agreement with the experimental values where (2.67% [21]≤δRe/Re≤4.53% [9]) and (0.07% [21]≤δωe/ωe≤1.8% [18]) yet these difference are 0.30% [5]≤δRe/Re≤5.24% [23], [26]) and (0.17% [5]≤δωe/ωe≤13.67% [22]) with respect to those theoretically determined. The calculated anharmonicity correction for the same state is ωexe = 2.09 cm-1 showing a relative difference of (2.8% [18]≤δωexe/ωexe≤30.6% [9]) with respect to values in literature. The ground state rotational constant Be=0.1030 cm-1 shows good accordance with 0.0967 cm-1 [18] (δBe/Be=6.51%) and 0.098 cm-1 [18] (δBe/Be=5.1%) which are obtained from the analysis of emission spectra and effective core potential calculations, respectively. The first excited state (1)2Π is unbound and purely dissociative correlating with the lowest asymptote. It interacts with the (2)2Π state at 2.05 Å where an avoided crossing changes the leading configuration and causes the abrupt change in their DMCs as shown in Figure 2. The PEC of the (2)2Π state describes a typical bound state whose calculated transition energy Te=34945 cm-1 is in excellent agreement with the experimental and theoretical values where the relative difference ranges between (2.34% [13]≤δTe/Te≤3.96% [10]). However less agreement exists between the calculated harmonic frequency ωe=322.6 cm-1 and the only value available in literature ωe=383 cm-1 [13]. The equilibrium distance Re=2.258 Å, which has not been reported before, is close to that of the ground state. With both being bound states having considerable dissociation energies, our attention has been drawn to further calculations of the Franck-Condon factors corresponding

to the X2Σ+_(2)2Π transition and so they are calculated and given in Table SM6 in the Supplementary Material. On the other hand, the X2Σ+_(2)2Σ+ transition has been studied extensively by means of experimental methods. It is reported that the B→X transition of diatomic mercury halides are somewhat complicated experimentally due to the significant separation between the equilibrium distances of the considered states [35]. The (2)2Σ+ transition energy with respect to the minimum of the ground state in this work is Te= 23707 cm-1, the equilibrium distance is Re =3.122 Å and the harmonic frequency is ωe=179.2 cm-1, all are in excellent agreement with the experimental values with which the percentage relative differences are (1.09% [18]≤δTe/Te≤1.22% [9], [21]), (3.38% [18]≤δRe/Re≤5.47% [21]) and (6.62% [18]≤δωe/ωe≤7.14% [9]), respectively, yet this agreement weakens upon comparison with the theoretical values reported by Wadt [22] where δTe/Te=7.67%, δRe/Re=6.55% and δωe/ωe=9.86%. The dissociation energy of this polarized state as indicated from the static dipole moment curve is reported to be 35811 cm-1 [5] from MRCI+Q calculations which is equivalent to the energy calculated by this work at 5Å where the dipole moment is |𝜇|=8.27 a.u. In fact, theoretical reproduction of the (2)2Σ+ state spectroscopic constants of mercury monohalides is a noteworthy task regarding the selection of basis sets. Several ECP basis sets describing 60 core electrons of mercury with variety of basis sets for the halogen have been tried with which the transition energy and harmonic frequency of this state were underestimated, e.g., Te~20000 cm-1 and ωe ~160 cm-1 for HgCl. To understand the origin of discrepancies with respect to the reported data, comparison is conducted among constants obtained from high level theoretical calculations, i.e., MRCI+Q [5] and CCSD(T) [24], [4]. The recent work of Saiz-Lopez et al. [5] in which a very similar level of theory is used, i.e., MRCI+Q, has employed an ECP of 60 core electrons to describe the Hg atom and aug-cc-pV(q+d)Z to describe all the chlorine electrons. Among the 37 valence electrons, 17 are active on 12 orbitals (Hg: 5d10, 6s2, 6p) and (Cl: 3p5). For the ground state, this approach has well predicted the equilibrium distance Re=2.324 Å but the harmonic frequency is significantly overestimated ωe = 351 cm-1 and the dissociation energy 𝐷00 = 7257 cm-1 is underestimated whereas the distance Re=2.96 Å of the (2)2Σ+ state is in very good agreement. However CCSD(T) calculations [4], [5] has shown better accordance considering the ground state constants. The lack of reliability of ECPs with 60 core electrons to estimate the harmonic frequency of the HgCl ground state together with producing reasonable transition energy for the B2Σ+ state might be linked to the electronic correlation effect of the mercuric 5d electrons and the inclusion of extra-diffuse functions within

the selected basis. To test that, we calculated the constants for the two states of interest X2Σ+ and B2Σ+ by reproducing the calculations above [5] but with changing the state of activity of the electrons in 5d orbital of Hg. Table 1 gives the constants as calculated upon keeping the 5d orbitals initially doubly occupied without being frozen in subsequent CI calculations (denoted by a2 entry) and then those orbitals are kept doubly occupied by the 10 frozen electrons (denoted by a3 entry). Though the constants estimated by the latter calculations for the ground state have shown better agreement with respect to the data available in literature, significant discrepancy is noticed regarding the transition energy of the excited state. Other than the ground state dissociation energy, the calculations (denoted by a1 entry), in which the correlation of the 5d10 electrons is represented by the ECP, have been able to reproduce the spectroscopic constants of the states being observed and reported experimentally showing reasonable agreements at least within the range of equilibrium distances. It is obvious that the calculated ground state dissociation energy improves upon the inclusion of the 5d orbitals of mercury in the active space and the diffuse functions in corresponding basis sets. Although diffuse functions are included within the augmented basis set of chlorine, ionic species require extra-diffuse functions appropriate for core and semi-core correlation [3]. Excited states are known to participate in photo-induced events as well as in thermally activated reactions, even in many cases in which only the ground state is believed to be involved [38]. Thus a wider view of the molecular spectrum showing as many states as could theoretical investigations provide may help to understand the interactions among these states especially when experiments involve external fields. Higher excited states are then investigated and Table 1 gives the corresponding constants as they are reported in this work for the first time in literature. The PECs and DMCs of six quartet states are presented in Figures 1 and 2 as well among which three states are bound; (1)4Σ+ lies at 44541 cm-1, (1)4Δ at 45394 cm-1 and (1)4- at 45862 cm-1. Theoretical calculations remain predictions until being compared with experimental data which is the case for the investigated doublet and quartet states other than those discussed above. 3.2 Static, permanent and transition dipole moments The electric dipole moment μ is one of the most fundamental properties with significant importance in several areas of research. For instance, it helps when studying the electron density, the electrostatic interactions of a molecule and the construction of the bonding models based on

molecular orbitals [39], [40], [41]. Moreover, it has been reported that the permanent dipole moment of the diatomic molecules plays an important role in the experimental sensitivity of eEDM experiments through the polarized field as well as in the quantum phases studies [2], [42], [43], [44]. The dipole moment operator is a simple sum of one-electron operators and its expectation value is related to the nature of valence electrons. Therefore, calculated values of dipole moment are considered among the most reliably predicted physical properties. As studies on the permanent electric dipole moment of the HgCl molecule are limited to the two lowest 2Σ+ states, we calculated the static dipole moment values and plotted them as function of the internuclear distance (Figure 2) then the permanent dipole moment of each bound state is determined at its equilibrium distance (Table 1). With this operator being dependent on the constructed wavefunctions which have well reproduced spectroscopic constants in excellent agreement with the experimental values, we may consider the values in this work accurate enough to represent the true physical system. Moreover, the transition dipole moment which is known to be a key value to determine Einstein coefficients and Franck-Condon Factors is computed for the transitions between the three lowest bound states and the corresponding curves are provided in Figure (SM1) in the supplementary material. By peering deep into the DMCs, the sign convention is defined such as the negative value corresponds to the Hgδ+Clδ- polarity with the mercury atom being at the origin. The ground state static dipole moment possesses negative values with the largest magnitude of |𝜇|=1.73 a.u. at R=2.74 Å and permanent value |𝜇𝑒|=1.37 a.u. at the equilibrium distance. The DMC of the (2)2 state among all Zn, Cd and Hg monohalides shows a characteristic behavior that describes a polarized state for which the negative values of the DMC sharply increase as it approaches the ionic dissociation limit Hg+(2S)+F-(1S) at large internuclear distances. The DMCs experience some abrupt gradient changes that is related to the occurrence of an avoided crossing between the PECs of two states of the same symmetry. At this region, two adiabatic solutions are given by mixing the wavefunctions of these states to avoid crossing of the corresponding energy curves. This interaction is reflected on the DMCs as a sharp slope changing indicating a reversed polarity of the atoms. For the provided PECs of doublet states, three main avoided crossing regions take place between (1)2(2)2 at 2.05 Å, (2)2-(3)2 at 2.02 Å and (3)2-(4)2 at 2.62 Å with energy separations of 1989 cm-1, 3706 cm-1 and 4170 cm-1, respectively. In addition, crossings among states of different symmetries also take place as the crossing between (2)2+ and (2)2 at 2.33 Å. Such behavior

leads to difficulty in accurate descriptions near the vicinity of these positions and to perturbations in the vibrational levels. 3.3 Molecular configurations and ionic character The molecular orbitals (MOs) of the HgCl molecule constructed from the atomic orbitals of Hg and Cl are (1 2 3 4 51 2 defining the active space that is to be filled with 9 valence electrons. The ionic configuration (12 22 14 31) dominates the ground state X2+ at short internuclear distances indicating a charge transfer from Hg to Cl. Its percentage weight is 56.5% at 2.2 Å, 55.2% at 2.33 Å, 51.4% at 2.68 Å. However the covalent configuration (12 21 14 32) is dominant at long distances. The percentage ionic character fionic=/eRe is also calculated to indicate the nature of a molecular bond such that 0≤fionic≤1, with f=1 being the ionic extreme. In Figure 4, the potential energies and the corresponding ionic characters of the four low lying states are plotted as function of R to show the regions where ionic or covalent bond nature is leading such that fionic bears large or nearly zero values, respectively. Within the equilibrium region the ground state bond is ionic 0.3≤fionic≤0.35 relative to elsewhere. Exclusively for the ground state, fionic is calculated using the Hg and Cl electronegativity values by applying different equations available in literature [45], [46] as given in Table 1. The curve corresponding to the (2)2+ state shows a counter behavior, i.e., covalent configuration is dominant at short distances while the ionic configuration leads to the dissociation into ionic fragments at long distances. An intersection between the curves of fionic of the two lowest 2Σ+ states at R=3.01 Å indicates a mixed (50%-50%) ionic-covalent nature of the bond after which bond nature dominance is flipped. This is justified by the interaction in the form of an avoided crossing between the two states. Similar analysis can be conducted for the two lowest 2Π states. At short distances, the ionic configuration (12 22 13 32) dominates the (1)2Π state with 70.6% percentage weight, then at 2.05 Å the charge remains at Cl with the molecule being in the (2)2Π state with 51% (12 22 14 21) at its equilibrium position. Table 1 gives the molecular configurations whose weights are greater than 5% for each bound state and its percentage ionic character at the equilibrium distance. These configurations illustrate how the excited molecular states arise from the promotion of electrons into the active molecular orbital space by single and double excitations. The occurrence of double excitations is associated with shallow energy potentials in which antibonding MOs are occupied.

3.4 Spin-orbit coupling effect Up to different percentages, spectroscopic properties of molecular states are affected by spin-orbit interactions that arise from the interaction of the electronic intrinsic magnetic moment with the orbital magnetic moment. In diatomic molecular calculations, the spin-free states (labeled 2s+1Λ± ) are identified according to their projection of the orbital angular momentum on the internuclear axis (Λ = 0, 1, 2, 3, 4). On the other hand, the spin-orbit electronic states (Ω states) are identified upon coupling the orbital angular momentum Λ with the spin angular momentum S through its projection Σ on the internuclear axis producing a total angular momentum Ω = |Λ±Σ| along the internuclear axis indicating the splitting of the parent states. Upon examining the percentage of each parent state at each energy point, the state symmetry and its spin angular momentum, electronic states with spin-orbit coupling effect are identified in the representation of Ω = 1/2, 3/2, 5/2, 7/2. In this work, the energies for the Ω molecular states are obtained by diagonalizing the total Hamiltonian Ht = He + WSO, where He is the Hamiltonian in the Born-Oppenheimer approximation for calculating the energies of the spin-free molecular states, and WSO is the spinorbit pseudopotential used to represent SO coupling. The ECPs representing the heavy atom (Hg) and the light atom (Cl) are both regarded to design the spin-orbit averaged semi-empirical pseudopotential to count for the spin-orbit effect on the molecular states of HgCl. Figures (4, SM2, SM3) show 17 states of Ω=1/2, 12 states of Ω=3/2, 6 states of Ω=5/2 and one state of Ω=7/2. Spectroscopic constants of the three lowest bound Ω-states and their dominant S parent compositions at Re are given in Table 2. Like spin-free states, crossing and avoided crossings also take place among states with spin-orbit effects. The crossing and avoided crossing points among Ω-states are given in Table SM1. Upon comparing the constants given in Table 1 for the spin-free states and those in Table 2 for the spin-orbit states, one can confirm the percentage of dominant parent composition for having nearly the same spectroscopic constants. The main S parent of (1)1/2, (3)1/2 and (2)3/2 states are 99.5% X2Σ+, 99.45% (2)2Σ+ and 99.72% (2)2Π states, respectively. These states are investigated and plotted, and their constants are reported in this work for the first time in literature. 3.5 Vibrational levels and Franck-Condon factors The points set of PECs corresponding to each bound state of the -S and Ω states is used as an input data file in LEVEL16 program [47] to solve the time-independent vibrational-rotational

Schrödinger equation and calculate the energy expectation values Ev of possible PEC levels (with J=0) and the first seven centrifugal distortion constants as well for each state. Then the abscissas of the turning points Rmin and Rmax are determined at each level. Table (SM2, SM3, SM4) within in the supplementary material gives Ev, Bv, Dv, Rmin, Rmax for many vibrational levels of the bound spin-free and spin-orbit states. As detailed in the preliminary introduction section above, laser spontaneous and stimulated emission spectra resulting from X2Σ+_(2)2Σ+ transition have been extensively studied all through their vibrational levels corresponding to strong intensities. Comparison among the energy values available in literature and those calculated in this work, given in Table 3, shows excellent agreement where the percentage relative energy difference ranges between (0.08%≤ %Δ𝑇𝜈′ ― 𝜈′′≤ 3.55%). To the best of our knowledge the vibrational structure of the spin-orbit electronic states in the HgCl molecule has not been reported before. Furthermore, upon calculating the transition dipole moment for X2Σ+-(2)2Σ+ and X2Σ+-(2)2Π transitions and based on the computed -S PECs, Franck-Condon Factors and radiative lifetimes of the discrete vibrational levels up to (𝜈′,𝜈′′) of (35, 35) and (20, 20), respectively, are obtained using the same program [31] by calculating the Einstein’s coefficient Aji which couples the two considered levels. FCFs and radiative lifetimes of these two transitions are given in Tables SM7 and SM8. These calculations are useful in describing the overlap of the vibrational wavefunctions for the spontaneous radiative transitions and in studying the diagonal transition probability between the low-lying excited states and the ground state. 4. Conclusion MRCI+Q calculations have been carried out in this work to have a wider picture of the electronic structure of the HgCl molecule by investigating the PECs and DMCs of 18 -S states at the spinfree level in the 2s+1± representation as well as 36 states while considering the spin-orbit coupling in the representation of Ω=1/2, 3/2, 5/2, 7/2. For the bound states, spectroscopic constants as Re, ωe, ωexe, Be, Te, e and 𝐷00 are computed. Also for the bound states, vibrational levels (with J=0) are investigated and several vibrational constants as Ev, Bv, Dv, Rmin and Rmin are determined. Transition dipole moments μTDM, Einstein coefficients 𝐴21 and Franck-Condon factors between the lowest states are also calculated. The molecular bond nature has been analyzed from different scopes verifying the covalent/ionic nature of the low-lying states at different internuclear distances. Comparison with experimental and theoretical data available in literature demonstrated significant

agreements and new data regarding different states is reported for the first time. Finally, we expect that the results of the present work would aid further experimental investigations on this molecule.

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Table 1: 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 (ωexe), the rotational constant (Be), the permanent

dipole moment (μe), the dissociation energy (𝐷00), the percentage ionic character (fionic), and the leading configuration at Re. Λ-state X2+

Re Te (Å) (cm-1) 2.331(a1) 0 2.430(a2) 2.464(a3) 2.230(b) 2.420(d) 2.395(e) 2.410(h) 2.364(j1) 2.392(j2) 2.460(k) 2.398(l) 2.46(m) 2.324(n1) 2.376(n2) 2.354(p)

ωe (cm-1) 293.6(a1) 275.6(a2) 282.3(a3) 292.6(b) 292.2(c) 298.97(d) 293.4(e, g) 285.3(h) 305.0(j1) 281.0(j2) 278.3(l) 262(m) 351(n1) 293.1(n2) 299.8(p)

ωexe Be -1 (cm ) (cm-1) 2.09(a1) 0.1030(a1) 2.01(a2) 0.0947(a2) 1.61(a3) 0.0922(a3) 1.60(b) 2.15(d) 1.77(e) 2.25(h)

| 𝜇 𝑒| (a.u.) 1.374(a1) 1.121(a2) 1.512(a3)

0.0967(d) 1.29(h) 0.098(h) 1.18(k)

𝐷00* fionic† (cm-1) 10165(a1) 0.31†(a1) 9310(a2) 0.24(a2) 10323(a3) 0.32(a3) 13231(b) 8350(d) 11944(e) 8902(h) 10108(j1) 7030(j2)

7257(n1) 8018(n2) 7964(p) 8600(q) 8953(r) 8526(s)

Configuration and weight (%)** at Re 1σ22σ21π44σ1 (55.2%), 1σ22σ21π32π13σ1 (14.3%), 1σ22σ11π43σ2 (9.6%), 1σ22σ11π43σ15σ1 (5.5%)

0.28(h) 0.26(k)

0.22(r)

1.07(t)

(2)2+

(4)2+

3.122(a1) 23707(a1) 2.995(a2) 22632(a2) 3.180(a3) 20632(a3) 23421(b) 23452(d) 3.02(d) (e) 23422(e) 2.96 22018(h) 2.93(h) 2.96(n1) 2.608(a1) 51644(a1)

179.2(a1) 203.8(a2) 160.6(a3) 192.0(b) 191.9(d) 192.3(e) 198.8(h)

0.46(a1) 0.0574 (a1) 2.118(a) 0.0624(a2) 0.0556(a3) 0.50(b) 0.48(d) 0.0621(d) 0.54(e) 0.51(h) 0.066(h) 2.13(h)

242.0(a1)

5.80(a1) 0.0823(a1)

0.903(a1) 2405(a1)

0.26‡(u) 0.25‡(v) 0.31‡(x) 0.36(a1)

0.38(h) 0.18(a1)

1σ22σ11π43σ2 (29.3%), 1σ22σ21π43σ1 (23.2%), 1σ22σ21π32π13σ1 (15.4%), 1σ22σ11π43σ14σ1 (13.7%), 1σ22σ21π44σ1 (7.4%) 1σ22σ11π43σ2 (39.2%), 1σ22σ21π32π13σ1 (18.4%), 1σ22σ21π43σ1 (9.8%), 1σ22σ11π43σ14σ1 (8.9%), 1σ22σ21π44σ1 (8.3%), 1σ22σ11π42π2 (7.5%)

(2)2

1.43(a1) 0.1072(a1)

0.999(a1) 18033(a1) 0.23(a1)

(2)2

2.285(a1) 34945(a1) 322.6(a1) 35782(c) 383(c) 35841(f) 36387(g) 2.801(a1) 59595(a1) 122.7(a1)

5.65(a1) 0.0713(a1)

1.014(a1) 606(a1)

0.19(a1)

(2)2-

2.845(a1) 60083(a1) 109.0(a1)

3.13(a1) 0.0691(a1)

1.008(a1) 895(a1)

0.19(a1)

(1)4+

2.830(a1) 44541(a1) 108.1(a1)

4.00(a1) 0.0699(a1)

0.734(a1) 677(a1)

0.14(a1)

(1)4

2.930(a1) 45394(a1) 84.3(a1)

4.54(a1) 0.0652(a1)

0.684(a1) 350(a1)

0.12(a1)

(1)4-

3.034(a1) 45862(a1) 63.7(a1)

3.11(a1) 0.0608(a1)

0.604(a1) 295(a1)

0.11(a1)

(a1) This

1σ22σ21π42π1 (51.0%), 1σ22σ11π42π13σ1 (25.2%), 1σ22σ21π32π2 (6.8%) 1σ22σ21π32π13σ1 (96.6%) 1σ22σ21π32π13σ1 (96.5%) 1σ22σ21π32π13σ1 (87.6%), 1σ22σ11π32π13σ2 (7.3%), 1σ22σ11π43σ14σ1 (5.2%) 1σ22σ21π32π13σ1 (92.3%), 1σ22σ11π32π13σ2 (7.6%) 1σ22σ21π32π13σ1 (92.5%), 1σ22σ11π32π13σ2 (7.5%)

work: MRCI+Q:Hg: ECP78MWB, Cl: ECP10MWB, [5a1, 2b1, 2b2, 0a2].

(a2, a3) This work: MRCI+Q: Hg: aug-cc-pVQZ-PP (60 electrons core), Cl: aug-cc-pV(Q+d)Z (all electrons), 37 valence electrons, [3a1, 2b1, 2b2, 0a2], 5d orbitals are kept doubly occupied, (a2): 5d electrons are not frozen in subsequent CI calculations, (a3): 5d electrons are frozen in CI calculations. (b)Ref.

[9] Exp: Analysis of emission spectra, (c)Ref. [13] Exp: Sensitized flash photolysis, (d)Ref. [18] Exp: Theoretical analysis of experimental emission spectra, (e)Ref. [21] Exp: Analysis fluorescence excitation spectra, (f)Ref. [10] Exp: Analysis of UV emission spectrum. (g)Ref. [11] Exp: Analysis of UV emission spectrum (h)Ref. [22] DZP ECP (j)Ref. [24] CCSD(T) using (j1) CRENBL, (j2) AVTZ-PP ECPs (k)Ref. [26] NESC/B3LYP (l)Ref. [25] QCISD/E60&G6 (m)Ref. [23] DFT (n)Ref. [5] (n1) MRCI+Q, (n2) CCSD(T) (p)Ref [4] CCSD(T) (q)Ref. [35] Exp: Theoretical analysis of experimental emission spectra (r)Ref. [27] Theo: Fitting empirical potential functions to RKR curves (s)Ref. [37] Exp: referring to the NIST-JANAF Thermochemical Tables (t)Ref. [44] Theo: LECCSD *Ref.

𝜔2𝑒

[34], 𝐷00 = 4 𝜔𝑒𝑥𝑒 ―

𝜔𝑒 2

+

𝜔𝑒𝑥𝑒 4

†By

applying 𝑓𝑖𝑜𝑛𝑖𝑐 = 𝜇/𝑒𝑅𝑒 ‡From electronegativity calculations χ = 3.16, χ F Hg = 1.9 [48] according to the formulas: 2 (u)Ref. [45], 𝑓 𝑖𝑜𝑛𝑖𝑐 = 0.16 (𝜒𝐹 ― 𝜒𝐻𝑔) +0.035 (𝜒𝐹 ― 𝜒𝐻𝑔) (v)Ref. [45], 𝑓 𝑖𝑜𝑛𝑖𝑐 = (𝜒𝐹 ― 𝜒𝐻𝑔)/(𝜒𝐹 + 𝜒𝐻𝑔) 3/2 (x)Ref. [46], 𝑓 /(𝜒𝐹.𝜒𝐻𝑔)3/4]) 𝑖𝑜𝑛𝑖𝑐 = 1 ― 𝐸𝑥𝑝 ( ― [(𝜒𝐹 ― 𝜒𝐻𝑔) **From calculations as described in (a1).

Only Weights larger than 5% are given.

Table 2: Spectroscopic parameters and permanent dipole moment μe of the lowest bound Ω states of HgCl at spin-orbit configuration interaction level of theory.

Ω State Te (cm-1) Re (Å) ωe (cm-1) Be (cm-1) |𝜇𝑒| (a.u) % (SΛ-parent) at Re Ref. (1) 1/2

0.00

2.326

296.2

0.1035

1.118

99.50 % X2Σ+

(a)

(3) 1/2

23766

3.121

179.1

0.0574

1.675

99.45 % (2)2Σ+

(a)

(2) 3/2

37229

2.282

319.7

0.1074

0.974

99.72 % (2)2Π

(a)

(a)

Present work.

Table 3: Comparison between experimental vibrational energies and corresponding values calculated in this work for HgCl molecule. Exp. Ref.

Ref. [16]

Ref. [17] Ref. [15] Ref. [14] (a) (b)

𝜈′

𝜈′′

-1 𝑇𝑒𝑥𝑝 𝜈′ ― 𝜈′′(cm )

-1 (a) 𝑇𝑡ℎ𝑒𝑜 𝜈′ ― 𝜈′′(cm )

%Δ𝑇𝜈′ ― 𝜈′′ (b)

8 1 8 1 4 10 3 7 2 1 0 6 8 0 0 1 0

27 20 30 22 25 32 25 30 24 23 22 29 31 21 22 23 22

18523.2 18523.2 18117.1 18117.1 18117.1 18117.1 17928.8 17928.8 17925.5 17925.5 17925.5 17903.1 17890.3 18127.0 17890.3 17922.3 17935.2

18131.1 18508.1 17526.6 18046.5 17889.8 17473.6 17721.2 17361.6 17769.7 17821.1 17875.4 17392.9 17334.1 18104.4 17875.4 17821.1 17875.4

2.12 0.08 3.26 0.39 1.25 3.55 1.16 3.16 0.87 0.58 0.28 2.85 3.11 0.12 0.08 0.56 0.33

This work %∆𝑇𝜈′ ― 𝜈′′ =

𝑐𝑎𝑙 𝑇𝑒𝑥𝑝 𝜈′ ― 𝜈 ― 𝑇𝜈′ ― 𝜈 ′′

′′

𝑇𝑒𝑥𝑝 𝜈′ ― 𝜈

′′

× 100

E (cm-1) 80000

(2)2-

(2)2(2)2 (4)2

70000

(1)2(3)2+ (1)2

60000

(1)4-

(2)4+

2 +

4 +

2

4

(3)2

(1)4 (1)4 (1)4+

(2)4

2

4

2 -

4 -

Hg (1Pu) + Cl (2Pu)

(4)2+

50000

Hg (3Pu) + Cl (2Pu) 40000

(2)2 30000

(2)2+ 20000

(1)2

Hg (1Sg) + Cl (2Pu)

10000

X2+ R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

Figure (1): Potential energy curves of doublet and quartet states of HgCl in the 2s+1± representation.

μ (a.u.) 2

(4)2 +

1

(3)2 + R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

-1 -2

X2 +

-3

(2)2 +

-4 -5 -6 2

(1)2 

1.5

(3)2 

1 0.5

(4)2 

R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

-0.5 -1

(2)2 

-1.5 -2 2

(2)2 -

1.8

(2)2 

1.6 1.4 1.2

(1)2 -

1

(1)2 

0.8 0.6 0.4 0.2

R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

2

(1)4

(1)4-

1.5

1

(1)4+

(2)4

0.5

(1)4

R (Å)

0 1.9 -0.5

-1

2.3

(2)4+

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

Figure (2): Static dipole moment curves of doublet and quartet states of HgCl in the 2s+1± representation. Solid lines,  = 1/2

Energy (cm-1) 70000 60000 50000 40000 30000 20000 10000

R (Å)

0 1.9

2.4

2.9

3.4

3.9

4.4

2.4

2.9

3.4

3.9

4.4

μ (a.u.) 2 1.5 1 0.5 0 1.9 -0.5

R (Å)

-1 -1.5

Figure (3): Potential energy curves and static dipole moment curves of HgCl molecular states in the representation of Ω=1/2.

% fionic

Energy (cm-1)

1.00 50000

0.90 0.80

40000

0.70

(2)2+

30000

0.60 0.50 0.40

20000

0.30

X2+

10000

0.20 0.10

0

0.00

1.9

2.4

2.9

3.4

3.9

4.4

4.9

5.4

R (Å)

% fionic

Energy (cm-1)

0.50 50000

0.45

(2)2

40000

0.40 0.35 0.30

30000

0.25 0.20

20000

(1)2

10000

0.15 0.10 0.05

0

0.00

1.9

2.4

2.9

3.4

Energy

3.9

4.4

4.9

5.4

R (Å)

% fionic fionic

Figure (4): Variation of percentage ionic character fionic in accordance with potential energy as function of internuclear distance.

Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states Authors: Soumaya Elmoussaoui1*, Wael Chmaisani1 1Plateforme

de Recherche et d’Analyse en Sciences de l’Environnement (PRASE), Ecole

Doctorale de Sciences et Technologie (EDST), Université Libanaise, Hadath, Liban.

CRediT author statement

Soumaya Elmoussaoui: Conceptualization, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing – Original Draft, Writing – Review and editing, Visualization.

Wael Chmaisani: Conceptualization, Software, Validation, Formal analysis, Resources, Data curation, Writing – Review and editing

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states Soumaya Elmoussaoui1*, Wael Chmaisani1 1Plateforme

de Recherche et d’Analyse en Sciences de l’Environnement (PRASE), Ecole

Doctorale de Sciences et Technologie (EDST), Université Libanaise, Hadath, Liban.

The electronic structure of the low-lying molecular states of the HgCl molecule is investigated at MRCI +Q level of theory. Potential energy curves for 18 -S states and 36 Ω states are plotted and then employed to determine molecular spectroscopic constants, rovibrational parameters, Franck-Condon Factors, and transition properties. The ionic character of the lowest states is also analyzed.

E (cm-1) 80000

(2)2-

(2)2(2)2 (4)2

70000

(1)2(3)2+ (1)2

60000

(1)4-

(2)4+

2 +

4 +

2

4

(3)2

(1)4 (1)4 (1)4+

(2)4

2

4

2 -

4 -

Hg (1Pu) + Cl (2Pu)

(4)2+

50000

Hg (3Pu) + Cl (2Pu) 40000

(2)2 30000

(2)2+ 20000

(1)2

Hg (1Sg) + Cl (2Pu)

10000

X2+ R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

Solid lines,  = 1/2

Energy (cm-1) 70000 60000 50000 40000 30000 20000 10000

R (Å)

0 1.9

2.4

2.9

3.4

3.9

4.4

2.4

2.9

3.4

3.9

4.4

μ (a.u.) 2 1.5 1 0.5 0 1.9 -0.5 -1 -1.5

R (Å)

% fionic

Energy (cm-1)

1.00 50000

0.90 0.80

40000

0.70

(2)2+

30000

0.60 0.50 0.40

20000

0.30

X2+

10000

0.20 0.10

0

0.00

1.9

2.4

2.9

3.4

3.9

4.4

4.9

5.4

R (Å)

% fionic

Energy (cm-1)

0.50 50000

0.45

(2)2

40000

0.40 0.35 0.30

30000

0.25 0.20

20000

(1)2

10000

0.15 0.10 0.05

0

0.00

1.9

2.4

2.9

3.4

Energy

3.9

4.4

4.9

% fionic fionic

5.4

R (Å)

Theoretical electronic structure and rovibrational calculations with spin-orbit effect of the HgCl low-lying electronic states Soumaya Elmoussaoui1*, Wael Chmaisani1 1Plateforme

de Recherche et d’Analyse en Sciences de l’Environnement (PRASE), Ecole

Doctorale de Sciences et Technologie (EDST), Université Libanaise, Hadath, Liban.

Highlights 

MRCI +Q calculations are carried out to investigate the electronic structure of the low lying states of the HgCl molecule.



18 states are investigated at the spin free level and 36 states considering the spin-orbit coupling effect.



Spectroscopic constants of bound states are calculated and the ionic character of the lowest states is analyzed.



Constants of vibrational levels of low lying -S and Ω bound states are computed.



Transition dipole moment calculations are computed and used to calculate Einstein spontaneous emission coefficients and radiative lifetimes between discrete vibrational levels.

E (cm-1) 80000

(2)2-

(2)2(2)2 (4)2

70000

(1)2(3)2+ (1)2

60000

(1)4-

(2)4+

2 +

4 +

2

4

(3)2

(1)4 (1)4 (1)4+

(2)4

2

4

2 -

4 -

Hg (1Pu) + Cl (2Pu)

(4)2+

50000

Hg (3Pu) + Cl (2Pu) 40000

(2)2 30000

(2)2+ 20000

(1)2

Hg (1Sg) + Cl (2Pu)

10000

X2+ R (Å)

0 1.9

2.3

2.7

3.1

3.5

3.9

4.3

4.7

5.1

5.5

Solid lines,  = 1/2

Energy (cm-1) 70000 60000 50000 40000 30000 20000 10000

R (Å)

0 1.9

2.4

2.9

3.4

3.9

4.4

2.4

2.9

3.4

3.9

4.4

μ (a.u.) 2 1.5 1 0.5 0 1.9 -0.5 -1 -1.5

R (Å)

fionic

Energy

1.00 50000

0.90 0.80

40000

0.70

(2)2+

30000

0.60 0.50 0.40

20000

0.30

X2+

10000

0.20 0.10

0

0.00 1.9

2.4

2.9

3.4

3.9

4.4

4.9

5.4

fionic

Energy

0.50 50000

0.45

(2)2

40000

0.40 0.35 0.30

30000

0.25 0.20

20000

(1)2

10000

0.15 0.10 0.05

0

0.00 1.9

2.4

2.9

3.4

Energy

3.9

4.4

fionic

4.9

5.4