Ab initio MRCI+Q Investigations of Spectroscopic Properties of Several Low-lying Electronic States of S₂⁺ Cation

Abstract

The complete active space self-consist field method followed by the internally contracted multireference configuration interaction method has been used to compute the potential energy curves of $X^2\prod_g$, $a^4\prod_u$, $A^2\prod_u$, $b^4\sum_{g}^{-}$, and $B^2\sum_{g}^{-}$ states of $S{_2}^+$ cation with large correlation-consistent basis sets. Utilizing the potential energy curves computed with different basis sets, the spectroscopic parameters of these states were evaluated. Finally, the transition dipole moment and the Franck-Condon factors of the transition from $A^2\prod_u$ to $X^2\prod_g$ were evaluated. The radiative lifetime of $A^2\prod_u$ is calculated to be 887 ns, which is in good agreement with experimental value of $805{\pm}10$ ns.

Introduction

Sulfur dimer (S2) and its cation (S2+) are important mole-cules in astrophysics, astrochemistry, and chemical lasers. For example, the spectra of S2 has been detected in cometary atmospheres,1,2 Jupiter’s atmosphere, 3 and dense molecular clouds.4 In addition, S2+ cation is always generated from all kinds of industrial and natural plasmas containing sulfur compounds.5-8 Since sulfur compounds play an important role in a variety of research fields, the studies of spectroscopic properties and electronic states of sulfur compounds1-4,9-12 have attracted much attention over many years. Compared with extensive investigations of sulfur compounds, there are a few studies on the spectroscopic and transition properties of S2+ cation.

Early in 1975, Berkowitz et al.13 observed the photo-electron spectra of S2 and Te2. Based on the single ionization spectrum of S2, they identified the X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g states of S2+. At the same time, Dyke et al.14 also observed the He(I) photoelectron spectra of S2, assigned the X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g states of S2+, and fitted the spectroscopic parameters of the five electronic states of S2+. Later on, Tsuji et al.5 made a vibrational analysis of the A2∏u-X2∏g transition in S2+. Subsequently, rotational analysis of the S2+ (A2∏u-X2∏g) emission band was reported by Capel et al.6 and Brabaharan et al..15 Recently, the A2∏u-X2∏g emission spectrum of S2+ was observed through microwave discharge of CS2 or sulfur vapor in solid neon,7 and photolysis of an H2S2/Ar matrix in solid argon,8 respec-tively.

However, to the best of our knowledge, only a few theore-tical studies were made to investigate the spectroscopic properties of S2+ cation. In 1989, the total energy and bond length of the ground state X2∏g of S2+ are calculated by Balaban et al.10 using ab initio method. Recently, Grant et al.11 investigated the electronic structure of the ground state for S2+ by employing the CCSD(T) theory. The potential energy curve (PEC) of the ground state for S2+ was extra-polated to the complete basis set (CBS) by utilizing syste-matic sequences of correlation-consistent basis sets with an exponential function. Nevertheless, the previously available theoretical calculations are not enough to illuminate the spectroscopic properties of low-lying excited electronic states of S2+ cation.

In the present study, we performed ab initio calculations on the low-lying electronic states of S2+. The core-valence correlation and scalar relativistic (mass-velocity and Darwin term) corrections were taken into account. The PECs of 5 Λ-S states (X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g) were cal-culated with high-level multireference methods. In order to eliminate errors due to the incomplete basis set, the PECs were computed with a series of correlation-consistent basis sets and extrapolated to the CBS limit. On the basis of PECs of the bound Λ–S and Ω electronic states, the spectroscopic constants of the bound states were determined by numeri-cally solving the nuclear-motion Schrödinger equations. Finally, the transition dipole moment (TDM) and the radia-tive lifetime of A2∏u were obtained. The spin-orbit coupling (SOC) effect was included in computations on transition properties of X2∏g-A2∏u.

Figure 1.The PECs of X2∏g and A2∏u states determined by the MRCI+Q/CBS+CV+DK calculations (top and right axes), and the transition dipole moment of A2∏u-X2∏g (bottom and left axes).

Methods and Computational Details. In the present work, the electronic structure computations were performed with MOLPRO 2010 quantum chemical package designed by Werner et al..16 The point group of the S2+ cation is D∞h. Nevertheless, owing to the limit of the MOLPRO procedure, all of the computations were carried out in the D2v subgroup of the D∞h point group. The correlating relationships for the irreducible representations of the D∞h and D2v are ∑+g = Ag, ∑−g = B1g, ∑+u = B1u, ∑−u = Au, ∏g = B2g + B3g, and ∏u = B2u + B3u. In the subsequent calculations, the PECs of X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g electronic states of S2+ were cal-culated through the complete active space self-consistent field (CASSCF) method.17,18 In the CASSCF computations, active space was made up of eight MOs: two Ag, one B3u, one B2u, two B1u, one B2g, one B3g symmetric MOs. The 3s3p valence electrons of S were placed into the active space. The other twenty electrons of S2+ were distributed into the closed orbitals, i.e., three Ag, one B3u, one B2u, three B1u, one B2g, and one B3g symmetric MOs, which correspond to inner-shell orbitals 1s2s2p of S. Furthermore, all configurations in the configuration interaction (CI) expansions of the CASSCF wave functions were used as reference for inter-nally contracted multireference configuration interaction method19 (MRCI) and MRCI with the Davidson correction (MRCI+Q).20 Additionally, the core-valence (CV) corre-lation induced by n = 2 orbital of S atom was estimated by combining the MRCI+Q method and the aug-cc-pwCVQZ basis set.21 The 1s core orbital of sulfur atom was excluded in CV computations. In order to improve the quality of spectroscopic constants, the scalar relativistic effect was taken into account via the second-order Douglas-Kroll-Hess (DKH) one-electron integrals in the PECs calculations. The scalar relativistic effect (denoted as DK) was produced by the difference between the energies with DKH and without DKH using an aug-cc-pVQZ-dk22 basis set at the MRCI+Q level. The sensitivity of calculated electronic states to the basis set was investigated by using a series of correlation consistent basis sets (aug-cc-pV(n + d)Z, n = Q(4), 5, 6).23 For the sake of brevity, the basis set is abbreviated to aVnZ. The dynamical correlation energy was extrapolated to the CBS limit by n−3 extrapolation formula24-27 with n = Q, 5.

The SOC was calculated by employing the state inter-action method with the full Breit-Pauli (BP) operator,28 which means that the spin-orbit eigenstates are determined by diagonalizing in the basis eigenfunctions of On the basis of PECs obtained by MRCI + Q/CBS + CV + DK + SOC level, we then solved the nuclear-motion Schrödinger equations utilizing the numerical integration LEVEL program29 designed by Le Roy to obtain the corre-sponding vibrational wave functions, vibrational energy levels, Franck-Condon factors (FCFs), and spectroscopic constants.

 

Results and Discussion

Spectroscopic Parameters. In order to obtain more accurate PECs, the point spacing interval of the calculated electronic states was 0.05 Å for R = 1.3-2.4 Å, 0.1 Å for R = 2.5-4.0 Å, and 0.5 Å for R = 4.5-6.0 Å. Table 1 lists the calculated parameters of S2+, including adiabatic transition energies Te, vibrational constants (ωe and ωexe), rotational constants Be, and equilibrium distances Re. In Table 1, the spectroscopic constants were evaluated with the MRCI method utilizing the AVQZ and aV5Z basis sets. Table 1 also lists the previously available experimental results. Compared with previous accurate experimental results,15 the ωe, ωexe, Be, and Re of X2∏g and A2∏u states evaluated with the AVQZ basis set are accurate with deviations of 50.961, 0.5905, 0.0453 cm−1, and 0.0131 Å, but the spectroscopic constants of the two states evaluated with the AV5Z basis set are accurate to within 9.65, 0.0768, 0.00275 cm−1, and 0.0114 Å, respectively. For A2∏u state, the Te value cal-culated with the AVQZ basis set differs from experimental result15 by 331.63 cm−1, and the Te value calculated with the AV5Z basis set is only 307.25 cm−1 larger than experimental data.15 On the whole, the spectroscopic parameters obtained by the AV5Z basis set are more accurate. Thus, we use the AV5Z basis set to calculate the Davidson correction (MRCI+Q). Table 1 also lists the spectroscopic parameters derived from the PECs including CV and DK effects. As shown in Table 1, the Te values of a4∏u, A2∏u, b4∑−g, and B2∑−g states obtained with the MRCI+Q method are 17817.16, 22013.82, 30773.25, and 38898.84 cm−1, respectively, which are 228.07, 638.88, 716.91, and 900.67 cm−1 smaller than those calculated by the MRCI method. When only the core-valence correlation correction is taken into account in the present MRCI+Q calculations, Te is increased by 157.00, 526.03, 464.99, and 547.14 cm−1 for a4∏u, A2∏u, b4∑−g, and B2∑−g states, respectively; ωe is increased by 6.79, 4.31, 7.14, 3.93, and 5.20 cm−1 for X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g states, respectively. When only the relativistic correction is taken into account in the present MRCI+Q calculations, the influence of DK correction on the spectroscopic constants is evidently smaller than that with the core-valence correlation correction. For example, the DK correction makes the values of Te shift only by 86.38, 116.74, 106.52, and 79.06 cm−1 for a4∏u, A2∏u, b4∑−g, and B2∑−g states, respectively. Even though the influence of the DK correction on spectroscopic cons-tants is relatively small, it cannot be omitted in high-level ab initio computations.

By incorporating the Davidson correction as well as the CV and DK corrections into the present study, we calculate the spectroscopic constants which agree very well with the previous experimental values.8,15 For instance, the differ-ences between our calculated Te of the A2∏u state and the experimental values8,15 are 107.58-333.03 cm−1. Regarding the vibrational frequencies of X2∏g and A2∏u, our calculated values of ωe and ωexe differ by less than 10.33 and 0.0461 cm−1 from the accurate experimental results.7,15 For the rotational constants of X2∏g and A2∏u, our calculated Be values differ by less than 0.00145 cm−1 from experimental results.15 Compared with the experimental values,14,15 the calculated results of Re of the two states are accurate, only with deviations of less than 0.0061 Å.

Table 1.Note: For the experimental values, the inaccuracy of measurement is depicted in the bracket. aReference 14. bReference 5. cReference 6. dReference 15. eReference 7. f Reference 8

Table 2 lists spectroscopic constants obtained from MRCI+Q PECs at aV5Z and CBS levels. Compared with spectro-scopic constants determined by the MRCI+Q/aV5Z calcu-lations, the extrapolation to the CBS limit (excluding the CV and DK effects) makes Te increase by 246.03, 230.27, 190.71, and 173.39 cm−1 for a4∏u, A2∏u, b4∑−g, and B2∑−g states, respectively. Compared with spectro-scopic constants determined at the MRCI+Q/aV5Z+CV+DK level, the extra-polation to the CBS limit makes Te increase by 558.21, 543.86, 503.4, and 489.04 cm−1, respectively, for a4∏u, A2∏u, b4∑−g, and B2∑−g states. As to the A2∏u state, the Te obtained by the MRCI+Q/CBS+CV+DK calculations is larger than previously available experimental data15 by 651.44 cm−1, the deviations of ωe, ωexe, Be, and Re determined at the MRCI+Q/CBS+CV+DK level from the experimental values14,15 are only 6.185 cm−1, 0.0503 cm−1, 0.00035 cm−1, and 0.0015 Å, respectively. As to the X2∏g state, the deviations of ωe, ωexe, Be, and Re determined by the MRCI+Q/CBS+CV+DK calculations from the experimental values14,15 are also only 2.651 cm−1, 0.014 cm−1, 0.000826 cm−1, and 0.0022 Å, respectively. Our calculated ωe, ωexe, and Re of X2∏g state are 808.75 cm−1, 3.4111 cm−1, and 1.8217 Å, respectively, which also agree well with those of previous theoretical values11 of 816.9 cm−1, 3.1 cm−1, 1.8240 Å. In comparison to previous experimental and theoretical results,14,15 we can conclude that the spectroscopic constants determined by the MRCI+Q/CBS+CV+DK calculations are more accurate, even if the spectroscopic constants obtained by the MRCI+Q/AV5Z+CV+DK calculation also agree well with experi-mental data. Additionally, the spectroscopic constants for a4∏u, b4∑−g, and B2∑−g states have not been measured in experiments. However, we believe that the spectroscopic constants of these states are also accurate owing to the good consistence with experimental results for the X2∏g and A2∏u states.

Table 2.Spectrcoscopic constants of low-lying electronic states of S2+ determined by MRCI+Q computations at the AV5Z and CBS

Effect of Spin-Orbit Coupling on PECs of X2∏g and A2∏u States. The SOC effect generally results in the splitt-ing of multiplet electronic states. The spin-orbit splitting of the X2∏g and A2∏u states has been determined experi-mentally6 utilizing a rotational analysis of the A-X emission band. Table 3 lists the spectroscopic constants of the X2∏g and A2∏u states determined by the MRCI+Q/CBS+CV+DK calculations including the SOC effect. The calculated spin-orbit splitting of the X2∏g and A2∏u states are 425.61 and 23.39 cm−1, respectively, which are in reasonable agreement with experimental data of 469.7 ± 2.3 and 13.5 ± 2.7 cm−1.6 For the X2∏g state, the modifications caused by the SOC effect are only 0.23 cm−1, −0.0063 cm−1, 0 cm−1, and 0 Å, for spectroscopic constants ωe, ωexe, Be, and Re, respectively. For the A2∏u state, the modifications caused by the SOC effect are only 0.14 cm−1, −0.0058 cm−1, 0 cm−1, 0 Å, and 55.89 cm−1 for these spectroscopic parameters, respectively. According to the above discussion, we can conclude that the SOC effect cannot bring obvious modification to the spectro-scopic parameters of X2∏g and A2∏u.

Table 3.Spectroscopic constants of the X2∏gi and A2∏ui states of S2+ including the spin-orbit coupling effect

Transition Dipole Moment and Radiative Lifetime of A2∏u State. The electronic transition dipole moment (TDM) function of the A2∏u-X2∏g transition was computed. For the sake of clarity, the TDM of A2∏u-X2∏g as a function of the internuclear distance and PECs of the two states are plotted in Figure 1. It is found from Figure 1. that the TDM increases monotonously as the internuclear distance increases from 1.35 to 6.0 Å, and equals 0.29 a. u. (1 a. u. = 2.542 Debye) at the equilibrium distance of the A2∏u state. On the basis of the PECs obtained by the MRCI+Q/CBS+CV+DK+SOC calculations, we evaluated the spectroscopic constants, vib-rational wave functions, and vibrational energy levels of X2∏g and A2∏u. The spectroscopic constants of the two states have been analyzed in above Section, and vibrational level G(ν), vibration-dependent rotational constant Bν of the first 11 vibrational states for the two states are listed in Table 4. For the X2∏g state, the largest deviation of G(ν) and Bν from experimental values15 are only 30.4023 cm−1 (0.38％ for ν = 10) and 0.0008 cm−1, respectively. For the A2∏u state, the largest deviation of G(ν) and Bν from experimental values15 are only 56.1679 cm−1 (1.03％ for ν = 10) and 0.0055 cm−1, respectively. On the whole, our calculated values of G(ν) and Bν of the X2∏g and A2∏u states agree well with the previous experimental results. Subsequently, we evaluated the Franck-Condon factors (FCFs) from the vibrational level ν = 0-6 of the upper electronic state (A2∏u) to the vibrational level ν′ = 0-6 of the lower ground state (X2∏g), as listed in Table 5. It is found that the maximum FCFs of ν′-ν″ transitions in A2∏u-X2∏g system are 6.67 × 10−2 (6-0), 9.45 × 10−2 (5-1), 1.02 × 10−1 (3-2), 1.07 × 10−1 (2-3), 1.26 × 10−1 (1-4), 1.40 × 10−1 (0-5), and 1.74 × 10−1 (0-6) for ν″ = 0-6 vibrational energy levels, respectively, which agree well with the corresponding experimental values: 7.24 × 10−2, 9.53 × 10−2, 1.02 × 10−1, 1.11 × 10−1, 1.05 × 10−1, 1.51 × 10−1, and 1.81 × 10−1.15

Table 4.aReference15.

Table 5.aReference 15.

The transition probability from excited state (A2∏u) to the ground state is equal to the Einstein coefficient . The Einstein coefficient Aν′ν″ for spontaneous emission between vibrational levels ν′ and ν″ is defined by

where is the transition energy in unit of cm−1, TDM is the average electronic transition dipole moment in Franck-Condon region in atomic unit, and qν′ν″ is the FCF between vibrational levels ν′ and ν″. The radiative lifetime of vibra-tional level ν′ is defined as the inverse of the total transition probability

On the basis of Eq. (2), FCFs, and TDM of A-X, the radiative lifetime of ν′ = 0 vibrational level of A2∏u is calculated to be 887 ns, which agrees well with the previous experimental value of 805 ± 10 ns measured in a solid argon matrix.

 

Conclusion

In the present paper, the PECs of the low-lying electronic states (X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g) for the S2+ cation were investigated by the MRCI method with the correlation-consistent basis sets (aug-cc-pV(n+d)Z, n = Q, 5, 6). The Davidson and the core-valence correlation corrections were also taken into account in calculations. Subsequently, on the basis of PECs obtained by the CASSCF and MRCI+Q method with different correlation-consistent basis sets, we obtained the PECs of X2∏g, a4∏u, A2∏u, b4∑−g, and B2∑−g states, which have been extrapolated to the CBS limit. Based on the computed PECs, the spectroscopic constants of the corresponding states were evaluated, which agree well with the existing experimental results. The spin-orbit coupling of X2∏g and A2∏u states was taken into account via state interaction method with the full Breit-Pauli Hamiltonian. The spin-orbit splittings of X2∏g and A2∏u states were found to be consistent with the experimental data. Utilizing the PECs determined by the MRCI+Q/CBS+CV+DK+SOC calculations, vibrational levels G(ν), vibration-dependent rotational constants Bν for each vibrational state of X2∏g, and A2∏u states were evaluated by solving nuclear Schrödinger equations. The transition dipole moment function of spinallowed transition A2∏u-X2∏g was investigated, and the radiative lifetime of A2∏u (ν′ = 0) vibrational level was evaluated. Our studies indicate that core-valence correlation, and relativistic corrections have great influence to the spectroscopic parameters of S2+. The present theoretical investigation should help to understand the transition and spectroscopic properties of the low-lying electronic states of the S2+ cation.