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# Physical Sciences Reviews

Ed. by Giamberini, Marta / Jastrzab, Renata / Liou, Juin J. / Luque, Rafael / Nawab, Yasir / Saha, Basudeb / Tylkowski, Bartosz / Xu, Chun-Ping / Cerruti, Pierfrancesco / Ambrogi, Veronica / Marturano, Valentina / Gulaczyk, Iwona

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Volume 2, Issue 9

# Theoretical study of the electronic states of newly detected dications. Case of MgS2+ AND SiN2+

Toufik Khairat
/ Mohammed Salah
/ Najia Komiha
Published Online: 2017-08-01 | DOI: https://doi.org/10.1515/psr-2016-5101

## Abstract

The dications MgS2+ and SiN2+, experimentally observed by mass spectroscopy, are theoretically studied here. The potential energy curves of the electronic states of the two dications MgS2+ and SiN2+ are mapped and their spectroscopic parameters determined by analysis of the electronic, vibrational and rotational wave functions obtained by using complete active space self-consistent field (CASSCF) calculations, followed by the internally contracted multi-reference configuration interaction (MRCI)+Q associated with the AV5Z correlation consistent atomic orbitals basis sets. In the following, besides the characterization of the potential energy curves, excitation and dissociation energies, spectroscopic constants and a double-ionization spectra of MgS and SiN are determined using the transition moments values and Franck–Condon factors. The electronic ground states of the two dications appear to be of X3nature for MgS2+ and X4 for SiN2+ and shows potential wells of about 1.20 eV and 1.40 eV, respectively. Several excited states of these doubly charged molecules also depicted here are slightly bound. The adiabatic double-ionization energies were deduced, at 21.4 eV and 18.4 eV, respectively, from the potential energy curves of the electronic ground states of the neutral and charged species. The neutral molecules, since involved, are also investigated here. From all these results, the experimental lines of the mass spectra of MgS and SiN could be partly assigned.

## 1 Introduction

In order to participate in forming a spectroscopic database of the species potentially existing in the interstellar medium, we look to determine the electronic structures, the potential energy curves (PECs) and the spectroscopic constants of species recently detected. In this context, we have studied here the MgS2+ and SiN2+dications detected in laboratory by Franzreb and Williams (Arizona State University) (see Figure 1) [1]. These species are shown to be long lived and metastable in the gas phase [2]. The theoretical study performed here is based on ab initio quantum chemistry calculations. The ab initio calculations, the theoretical determination of electronic structures, the potential energy curves and spectroscopic constants are definitely helpful to confirm the experimental results obtained by mass spectroscopy. We first investigate the neutral MgS and SiN molecules, and then we looked at the dications. The accurate ab initio calculations performed on the low-lying electronic states show the existence of, at least, 12 bound states of MgS and five bound states for MgS2+, six electronic states for the neutral SiN and six for its dication. The potential energy curves of all these states are mapped, and the spectroscopic constants, the transition moments and the double photoionization spectra are determined. A comparison with experiment is made and an investigation on the spectroscopy and dissociation dynamics as well. Some spectroscopic constants and vibrational levels of the neutral electronic states (Tables 1 and 2) are also calculated here.

Figure 1:

Mass spectrum of MgS2+ detected at half integer m/z 28. With the permission of authors of Ref [1].

## 2 Computational details

Ab initio calculations of the potential energy curves of the different electronic states are performed using the CASSCF method (complete active space self-consistent field) in which all possible electronic excitations resulting for the distributions of the valence electrons into the valence molecular orbitals (MOs) were allowed. These valence MOs, calculated in the C2v symmetry group, were correlated and all the electronic states of the same spin multiplicity were averaged together in the CASSCF calculations. These first CASSCF calculations are for determining the most important electronic states of the neutrals and the dications. The extended AV5Z basis sets used for Mg, S, Si and N atoms include diffuse orbitals in order to describe properly the upper electronic states supposed to be of Rydberg character. The augmented cc-pV5Z basis set constituted of orbitals of s,p,d,f,g,h, type could be considered as complete. These calculations were then followed by the MRCI ones (internally contracted multi-reference configuration interaction) [3] with AV5Z quality of atomic orbital basis sets as implemented in the Molpro package [4]. The inner MOs were frozen in these MRCI calculations, and all the valence MOs are included. The electronic states correlating to lowest asymptotes of MgS and SiN and their respective dications are determined. For MRCI calculations, the references are all configurations of the CI expansion of the CASSCF wave functions. In these reference spaces, the total number of configuration functions is 328 meaning 240 configurations for MgS and for SiN 254 contracted Gaussian type orbitals (cGTOs) are to be taken into account; this implies more than one million of contracted configurations in the whole CI calculations. The nuclear motion problem was then solved using the derivatives of the potential curves near the minimum energy distances. Standard perturbation calculations of the Cooley method were performed [5] and variational treatments [6] as well. The spectroscopic data of the bound states were then deduced. The potential energy curves of the lowest electronic states of the two neutral molecules and their dications have been used to calculate the Franck–Condon (FC) factors and simulate the double-photoionization spectra of MgS and SiN. The Level program [7] allows such a simulation. The methods used here are supposed to be accurate enough for the description of the relative energies of the computed electronic states.

## 3.1 Neutral MgS

Figure 2 displays the global potential energy curves of low-lying singlet and triplet electronic states of the MgS molecule. These electronic states are correlated to the four lowest dissociation asymptotes: Mg (3Pu) + S (3Pg), Mg (1Sg) + S(1Dg), Mg (3Pu) + S (3Pg) and Mg (1Sg) + S (3Pg). The energy ordering of the asymptotes is deduced from the literature [8] and compared to the values from our calculations. As shown by V.W. Ribas et al. [9] (Table 3), the ground state is of X1Σ+ symmetry and presents a multiconfigurational character with an important contribution of the two configurations, 7σ224 and 7σ2114 [9, 10, 11, 12, 13, 14]. The energies of the vibrational and rotational levels are obtained when solving the radial equation; the Numerov and Cooley methods [5] were used for that (see Table 2). Table 1 gives the spectroscopic constants of the bound electronic states of MgS, and we provide the equilibrium distances (bohr), the harmonic wave numbers, the anharmonic terms and the rotational constants and compare them to experimental values. Our results are very close to those of the literature and hence prove the reliability of our theoretical calculations. The theoretical vibrational spectrum could be mapped using the MRCI PECs. Figure 3 presents the spectrum obtained by using the transition moments between the ground state X1Σ+and 11Π, 21Π states (the transition moments between sigma states could not be calculated). According to this spectrum, the most probable transition is for 11Π state, the energy related for this transition is between 3.89 × 103 cm−1 and 15.79 × 103 cm−1. The FC factors for the different transitions between the vibrational levels of ground state and those of the 1Σ+ and 1Π bound states have been also used to simulate the vibrational spectrum (Figure 4). The transitions were fitted by Gaussian functions, and the line intensities were estimated from the FC factors by adopting a Boltzmann distribution to simulate the population of each rotational level at 300K. The spectrum depicted is dominated by transitions involving the ground state X1Σ+ to 11Π, 21Σ, 2 1Π, but the most significant transition is for 21Σ+ vibrational levels (the most intense peak), and the energy related for this is between 2.103 cm−1 and 104 cm−1 .The equilibrium distance Re (X1Σ+) = 4.103 bohr is relatively close to Re (21Σ+) = 4.20 bohr; hence the most probable transition is (ν’, ν’’) = (0,0).

Figure 2:

The potential curves of the neutral MgS. Singlet (lower panel) and triplet states (upper panel) are separately represented.

Figure 3:

The transition momenta based simulated vibrational spectrum of MgS.

Figure 4:

F.C factors based spectrum. FC factors versus ΔE of MgS.

Table 1:

MRCI spectroscopic constants of electronic states of MgS, including the harmonic wavenumber (ωe), the anharmonic terms (ωexe,ωeye), the rotational constants (βe, αe,γ). All values are in cm−1. Equilibrium distances (Re, in bohr), Te (in eV) correspond to the MRCI adiabatic excitation energy, De (in eV) is calculated as the difference between the energy of the electronic state at equilibrium and the energy at the maximum of the potential barrier.

Table 2:

Vibrational levels of the different electronic states of MgS. All values are in cm−1.

Table 3:

Energies of the three lowest dissociation limits of MgS.

## 3.2 MgS2+ dication

According to our theoretical calculations, the ground state X3Σ of the MgS2+ dication has the following electronic configuration (7σ)2(8σ)2(3π)2 obtained after the removal of the two electrons from the outermost 3π-orbital of the neutral molecule (7σ)2(8σ)2(3π)4 which is of X1Σ+ symmetry [9]. The equilibrium distance of the ground state of the dication is Re = 4.8 bohr, and this state presents a relatively deep potential well to the dissociation channel. The dissociation energy found is 1.20 eV (see Table 5). Figure 5 depicts the MRCI potential energy curves of singlet, triplet and quintet lowest states correlated to the six lowest dissociation asymptotes: Mg+ (2Sg) + S+ (4Su), Mg+ (2Sg) + S+ (2Du),Mg+(2Sg) + S+ (2Pu), Mg+ (2Pu) + S+ (4Su), Mg+ (2Pu) + S+ (2Du) and Mg+ (2Pu) + S+ (2Pu). The energy ordering of the asymptotes is deduced from the literature [8] and compared to the values from our calculations; they are found in good agreement for the first four asymptotes and with a little difference for the Mg+ (2Pu) + S+ (2Du) and Mg+ (2Pu) + S+ (2Pu) asymptotes (see Table 4). Table 5 lists the dominant configurations in the wave functions of the electronic states quoted at the equilibrium distance of the ground state X3Σ (4.8 bohr). The five electronic bound states are X3Σ, 11Δ, 13Π, 11Σ+, 11Π three states with very shallow potential well 23Σ, 21Δ and 21Σ+, 11 electronic dissociative states 15Σ, 33Σ, 13Δ, 23Δ,33Δ,41Δ, 13Σ+, 23Σ+, 31Π, 23Π, 33Π are also shown in Figure 5. On the other hand, some crossings of the PECs could be observed: 11Δ with 13Π, 11Σ+ with 11Π, 23Σ with 13Π but the more important crossings are found for the 15Σ, cutting all the other bound states; hence their ro-vibrational levels should be perturbed. The proximity of the electronic states induces interactions and numerous couplings between the angular momenta that produce avoided crossings between states of the same symmetry and spin multiplicity as 13Σ+and 23Σ+, 11Δ and 21Δ, 23Σand 33Σ, 23Π and 33Π, 11Σ+and 21Σ+,23Δ and 33Δ, 21Π and 31Π; in consequence, the analysis of the vibrational spectrum is complicated. The energies of the vibrational and rotational levels are obtained when solving the radial equation. The Numerov and Cooley methods [5] were used for that (Table 6). Table 7 gives the spectroscopic constants of the bound electronic states of MgS2+. We provide the equilibrium distances (bohr), the harmonic wavenumbers (cm−1), the anharmonic terms and the rotational constants (Be and ωex). We have calculated the FC factors for the different transitions: from the vibrational levels of X1Σ+ of the neutral to other vibrational levels of the following states of the dication: X3Σ, 11Σ+, 11Π, 13Π. FC factors have been used to simulate the vibrational double-ionization spectrum in order to obtain a fingerprint of this dication and to facilitate its detection and to help future experimental spectroscopic works on MgS2+ (Figure 6). Only transitions with DJ = 0 were considered. This approach, where the FC factors are calculated for the double-ionization spectrum simulation, was widely discussed and validated in similar works [15, 16, 17].

The first “band” obtained corresponds to the transition: $Mg{S}^{2+}\left({X}^{3}{\Sigma }^{-},v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2eMgS\left({X}^{1}{\Sigma }^{+},v=0\right)+hv.$$Mg{S}^{2+}\left({X}^{3}{\Sigma }^{-},v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2e---->MgS\left({X}^{1}{\Sigma }^{+},v=0\right)+hv.$

This transition appears with a significant intensity due to favourable FC factors. The shapes of the potential energy wells of these electronic states are in favour of a good overlap of the vibrational levels. The spectrum is represented in Figure 6 and an assignment of the transitions is given in agreement with the experimental mass spectrum of Figure 1. All the transitions have a significant intensity except for 11Π. An explanation is that the Re (11Π) = 4.9 bohr, very different from Re (X1Σ+) = 4.10 bohr of the neutral ground state; consequently, the transitions are between vibrational states of X1Σ+ and those of the continuum of 11Π with unfavourable overlap (see Figure 6), leading to weak intensities. The four peaks of the experimental spectrum are so assigned; the four electronic states of MgS2+ involved are hence X3Σ, 11Σ+, 11Π and 13Π.

Figure 5:

The potential energy curves of MgS2+states (each symmetry is also represented separately).

Figure 6:

FC factors based on double-ionization simulated vibrational spectrum. FC factors versus ΔE of MgS2+.

Table 4:

Energies of the four lowest dissociation limits of MgS2+.

Table 5:

Equilibrium distances (Re, in bohr) and dominant electronic configurations of the MgS2+ electronic states quoted for R = 4.8 bohr, correlating to the five lowest dissociation limits. Te (in eV) corresponds to the MRCI adiabatic excitation energy. We give, between brackets, the leading CI coefficient in the CASSCF wavefunction. De (in eV) is calculated as the difference between the energy of the electronic state at equilibrium and the dissociation energy.

Table 6:

Vibrational levels of the different electronic states of MgS2+ and those of the ground state of MgS. All values are in cm−1.

Table 7:

MRCI spectroscopic constants of the ground electronic state of MgS and those of the electronic states of MgS2+ including the harmonic wavenumber (ωe), the anharmonic terms (ωexe, ωeye), the rotational constants (βee). All values are in cm−1.

## 3.3.1 Determination of the low-lying electronic states of SiN2+

The low-lying doublet and quartet electronic states of SiN2+and its neutral parent molecule computed at the MRCI/aug-ccpV5Z level are mapped Figures X.7 and X.8, respectively. The minimum of the electronic ground state X2Σ+ of the neutral SiN is taken as the reference energy. All the electronic states correlating to the four lowest dissociation limits (i. e. Si+(2Pu)+ N+ (3Pg), Si2+ (1Sg)+ N (4Su), Si+ (2Pu)+ N+ (1Dg) and Si+ (2Pu)+ N+ (1Sg)) are calculated. The energy of these asymptotes and the correlated electronic states are given in Table 8. These asymptotes are not represented in the figures for more clarity. The X4Σ-electronic ground state of SiN2+ corresponds to the removal of two electrons from the 2π MOs of the electronic ground state of the parent neutral molecule. Its potential energy curve shows a well of about 1.40 eV. The adiabatic double-ionization energy of SiN is found at 18.45eV, calculated as the energy difference between the minima of the ground states of the neutral and that of the dication. At least, six electronic excited states of SiN2+are bound and show potential wells of several meV. In Table 9, the dominant electronic configurations of these states, together with the equilibrium distances, are given. The respective dissociation limits of the excited states are found lower in energy than their potential wells, proving the metastable character of these states. At 35 eV above the neutral ground state, the calculated electronic states are close in energy and can lead to interactions through vibronic or spin–orbit couplings and to the mixing of their electronic wave functions. Figure 7 shows the avoided crossings between the 12Π and the 22Π for RSiN at about 3.6 bohr, between 12Σ+and 22Σ+at 5.8 bohr and also between the 12Δ and the 22Δ for RSiN at more than 6 bohr. These avoided crossings lead to the formation of local minima in the upper states.

Figure 7:

MRCI/aug-cc-pV5Z potential energy curves of the lowest states of SiN2+. (short (left panel) and long range (right panel).

Figure 8:

Potential energy curves of the electronic states of SiN (left panel) and ground states of the SiN and SiN2+ species (right panel) with the close lying doublet and quartet states of the dication.

Table 8:

Energies of the lowest dissociation limits of SiN2+, evaluated using the dissociation energy of the electronic ground state of the neutral SiN and an adequate thermo-chemistry cycle. The nature of the resulting electronic states is also reported.

Table 9:

Dominant configurations and equilibrium distances (Re in bohr) of the bound electronic states of SiN2+.

Table 10 gives the equilibrium distances (Re in bohr), the rotational constants (Be in cm−1), the harmonic wavenumbers (ωe in cm−1) and the anharmonic term (ωexe in cm−1) deduced from the MRCI potential energy curves and the resolution of the nuclear motion calculations.

As no experimental data are known for this species, all these values are predictive. For the X4Σ-electronic ground state of this dication, the equilibrium distance is found at 3.936 bohr. This value is clearly larger than the one computed at 2.985 bohr for the electronic ground state X2Σ+ of the neutral SiN, resulting in the depletion of the 2π bonding MOs. As the bond length of the dication is longer than that of the neutral, the equilibrium rotational constant of the electronic ground state of the dication is computed at 0.41605 cm−1, smaller than the one of 0.72305 cm−1 computed for the ground state of the neutral molecule. Concerning the vibration constants of the dication, the ground state values of ωe and ωexe are computed as 390.305 cm−1 and 1.51086 cm−1, respectively. For the neutral molecule, these values are computed at 1142.82 cm−1 and 6.37467 cm−1, respectively. All the values computed for the X2Σ+ state of the neutral molecule are in close agreement with the experimental determination (Re = 2.985 bohr, Be = 0.723 cm−1, ωe= 1142.82 cm−1 and wexe= 6.37 cm−1 Table 11). A similar accuracy is expected for the values predicted for the electronic states of the dication.

Table 10:

Main spectroscopic constants including the equilibrium distances (Re in bohr), the equilibrium rotational constant (Be in cm−1), αe, γ; the harmonic and anharmonic wave numbers (ωe and ωexe in cm−1) of the bound electronic states of SiN2+calculated at the MRCI/AV5Z level.

Table 11:

Main spectroscopic constants including the equilibrium distances, the equilibrium rotational constants (Be in cm−1), the rotational constants (βe, αe, γ). All values are in cm−1, equilibrium distances (Re, in bohr), Te (in eV) corresponds to the MRCI adiabatic excitation energy, De (in eV) is calculated as the difference between the energy of the electronic state at equilibrium and the energy at the maximum of the potential barrier, the harmonic and anharmonic wave numbers (ωe and ωexe, ωeye in cm−1) of the bound electronic states of SiN.

## 3.3.2 Simulation of the double-ionization spectrum of SiN

To obtain a fingerprint of this dication in order to facilitate its detection and to help future experimental spectroscopic works on SiN2+, the double-ionization spectrum of SiN (Figure 9) has been simulated. The FC factors between the electronic ground state X2+ of the neutral SiN molecule and the electronic states of SiN2+ have been computed using the LEVEL program [7]. Only transitions with DJ = 0 were considered. The first “band” obtained corresponds to the transition: $Si{N}^{2+}\left({X}^{4}{\Sigma }^{+},v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2{e}^{-}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}SiN\left({X}^{2}{\Sigma }^{+},v=0\right)+hv.$$Si{N}^{2+}\left({X}^{4}{\Sigma }^{+},v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2{e}^{-}------->,SiN\left({X}^{2}{\Sigma }^{+},v=0\right)+hv.$

This transition appears with a low intensity due to unfavourable FC factors. It could not be represented in our theoretical spectrum. The deepness and width of the potential well of this electronic state of the dication and the difference of equilibrium distance between the ground states of the neutral and the dication are not in favour of this transition. Similar situation is obtained for the excited states of the dication possessing low barrier heights and larger equilibrium bond distances compared to the electronic ground state of the neutral SiN (i. e. the 12Π, 22Σ+, 12Σ, 42Π, 14Σ+,14Δ, 14Π, 14Σ and 24Δ). The only two allowed transitions, with favoured FC factors, spin and symmetry, are between doublets. The transition SiN2+(22Σ+) SiN(X2Σ+) is the most intense. The spectrum is represented in Figure 9 and an assignment of the transitions is given in agreement with the experimental mass spectrum of Figure 10. We should say that we have computed two transitions with favoured FC factors that are: $Si{N}^{2+}\left({1}^{4}{\Sigma }^{-},v=\phantom{\rule{0ex}{0ex}}0\right)+2eSiN\left({X}^{2}{\Sigma }^{+},v=0\right)and$$Si{N}^{2+}\left({1}^{4}{\Sigma }^{-},v=\phantom{\rule{0ex}{0ex}}0\right)+2e----->SiN\left({X}^{2}{\Sigma }^{+},v=0\right)and$$Si{N}^{2+}\left({1}^{4}\Pi ,v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2eSiN\left({X}^{2}{\Sigma }^{+},v=0\right).$$Si{N}^{2+}\left({1}^{4}\Pi ,v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0\right)+2e----->SiN\left({X}^{2}{\Sigma }^{+},v=0\right).$

As these transitions are not spin or symmetry allowed, they are not observed in the experimental mass spectrum.

Figure 9:

Franck–Condon factor-based simulation of the double-ionization spectrum of SiN.

Figure 10:

Experimental mass spectrum from reference[1] with the permission of the authors.

## 4 Conclusion

In the present theoretical work, a large study and a global view of the potential energy curves, transition moment functions, dissociation energies and the spectroscopic constants of the bound electronic states of MgS and SiN and their respective dications MgS2+ and SiN2+ have been performed. Electronic ground states and excited electronic states have been accurately described here. Our calculations predict the existence of metastable electronic states of MgS2+ and SiN2+. Most lines of the experimental mass spectra of these systems could be so assigned. The electronic states of the MgS2+ and SiN2+dications have been determined using highly accurate ab initio computations. The Potential Energy Curves (PECs) were calculated with significant potential wells proving the possible existence of such species. The electronic ground state of SiN2+is found of X4-nature (X3Σ for MgS2+) and presents a potential well of about 1.40eV (1.20 eV for MgS2+). Other excited electronic states are also predicted to be bound, and several couplings between these states are supposed to occur. The rotational and the vibrational spectroscopic constants of the bound states are computed. The FC double-ionization spectra of SiN and MgS are simulated using the potential energy curves of the bound states of the dications and those of the electronic ground states of the neutral molecules. The SiN spectrum is mainly composed by the contribution of two excited states of the dication (the 42Π and 22+), whereas the peaks due to the ground state and the other bound excited states appear with relatively low intensities, because of the large difference of the equilibrium bond length of these states and that of the ground state of the neutral molecule. The only allowed transitions are first between the X2+ state of the neutral and the 12+ state of the dication, and second, much less intense, between the X2+ state of the neutral and the 12Π state of the dication. The other double-ionization transitions are spin or symmetry forbidden. This simulated spectrum (Figure 9) is in agreement with the experimental mass spectrum produced at Arizona State University by Franzreb and Williams [1] (Figure 10). On the other hand, similarly, we could assign the four peaks of the experimental double-ionization spectrum of MgS. They are due to the transitions between the ground state of the neutral X1+ and the X3Σ, 11Σ+, 11Π, 13Π states of the dication. In this work, we could assign the observed bands in the experimental mass spectra of SiN and MgS. All the data given here could also be helpful to future experimental works dealing with the spectroscopy of these dications.

## Correction Statement

The DOI of this article has been used for another publication by mistake. If you intended to access the other publication, please use this link: https://doi.org/10.1515/psr-2016-0101

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Published Online: 2017-08-01

This research was supported by the Moroccan Research Program under the reference grant n° PU-SCH09/09 of the Ministry of Higher Education and a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Program under grant n° PIRSES-GA-2012-31754.

We thank Klaus Franzreb and Peter Williams for their permission to include their unpublished experimental data in our theoretical work.

Citation Information: Physical Sciences Reviews, Volume 2, Issue 9, 20165101, ISSN (Online) 2365-659X, ISSN (Print) 2365-6581,

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