The article information
 En Cao, Shuang Liu, Yuzhi Song
 曹恩, 刘爽, 宋玉志
 Accurate Theoretical Study of LiS Radical and Its Singly Charged Cation and Anion in their Ground Electronic State
 LiS分子和离子基态的精确理论研究
 Chinese Journal of Chemical Physics, 2017, 30(2): 128134
 化学物理学报, 2017, 30(2): 128134
 http://dx.doi.org/10.1063/16740068/30/cjcp1611219

Article history
 Received on: November 17, 2016
 Accepted on: January 20, 2017
Metal oxide species play a major role in a variety of environments, including catalysis, high temperature reactions, and stellar atmospheres [1,2], which have received an appreciable amount of studies [3]. The alkali metal monosulfide, LiS, was addressed by Partridge et al. [4] who deduced that the ^{2}∏ state is the ground state of LiS. By analyzing the observed millimeter/submillimeter spectrum, Brewster and Ziurys [5] obtained spectroscopic constants and equilibrium bond length in the ground state of LiS (X^{2}∏). Lee and Wright [6] calculated a set of potential energy curves (PECs) of LiS which were then extrapolated to complete basis set (CBS) using the twopoint extrapolation formula developed by Helgaker et al. [7,8]. By employing LeRoy's LEVEL 7.5 program [9], the spectroscopic constants were also calculated. By using the full valence complete active selfconsistent field (CASSCF) method [10] which was then followed by the internally contracted multireference configuration interaction approach (MRCI) [11,12], Khadri et al. [13] carried out detailed studies of LiS^{+} with the correlationconsistent basis sets of Dunning, ccPV5Z [14,15]. They obtained the spectroscopic constants of the lowest electronic states of LiS^{+}, including the harmonic and anharmonic vibrational frequencies, rotational constants, and dissociation energies. Boldyrev et al. [16] reported a study of the electronic ground state of LiS^{−}(^{1}Σ^{+}) with polarized spiltvalence basis sets (6311+G^{*}) [1720] at correlated secondorder (MP2) levels and selfconsistent field method (SCF). The equilibrium geometries were used to evaluate electron correlation in the frozencore approximation by full fourth order [21] mollerplesset perturbation theory and the (U) QCISD (T) method [22] for LiM^{n} (n=+1, 0, 1; M=Li, Be, P, C, N, O, F).
In order to obtain the accurate PECs of LiS^{n} (n=1, 0, +1), by employing the MRCI (Q) method [23], we carried out detailed studies of LiS^{n} (n=1, 0, +1) systems with the standard ugmented correlationconsistent basis sets, augccPV (T+d) Z (AVTdZ) and augccPV (Q+d) Z (AVQdZ). The corevalence (CV) correlation is carried out with the CV basis set augccPCVQZ, and the relativistic correction is also taken into account by using the augccPVQZDK basis set. In particular, the uniform singletpair and tripletpair extrapolation (USTE) protocol [2426] is employed to extrapolate the PECs calculated at AVXdZ (X=T, Q) to the CBS limit. All the PECs are then fitted to analytical potential energy function (APEFs) by using the formalism developed by Aguado and Paniagua [27,28]. By numerically solving the radical Schrodinger equation of the nuclear motion, we obtained vibrational levels, classical turning points, rotation and centrifugal distortion constants.
Ⅱ. COMPUTATIONAL DETAIL A. Ab initio calculationsThe ab initio calculations are carried out by using MOLPRO 2012 program [29]. In order to obtain the highlevel PECs of LiS^{n} (n=1, 0, 1), the potential energies are calculated for the internuclear separation ranging from 0.5 a_{0} to 35 a_{0} with the interval of 0.05 a_{0}, which declines to 0.01 a_{0} in the vicinity of the equilibrium geometries. All calculations are carried out at the MRCI (Q) level using the CASSCF wave function [1012] as the reference, which has been applied to many diatomic molecules [3032]. In the ab initio calculations of PECs of LiS^{n} (n=1, 0, 1), the AVXdZ (X=T, Q) atomic basis sets of Dunning [14,15] is employed for Li atom. For sulfur atom, the AVXdZ (X=T, Q) basis set is chosen, which includes highexpenent corepolarization d functions as performed for the second row atoms. Both the corevalence correction and the relativistic effect are considered. The corevalence (CV) correction is taken into account with the CV basis set augccPCVQZ, and the relativistic correction is carried out with the augccPVQZDK basis set, respectively. We employ C_{2v} point group symmetry in the ab initio calculations, which includes four irreducible representations, namely A_{1}, B_{1}, B_{2} and A_{2}, respectively. For LiS^{n} (n=1, 0, 1), 8 orbitals (4a_{1}+2b_{1}+2b_{2}) are confirmed as the active space. Thus, in order to select molecular state, we choose the A_{1} irreducible representation of the C_{2v} point group and carry out the twostate average calculation.
B. Extrapolation to CBS limitThe MRCI (Q) electronic energy can be treated in spilt form, which can be written as [33]
(1) 
where the subscript X indicates that the energy has been calculated in the AVXdZ (X=T, Q) basis sets, while the superscripts dc and CAS stand for the dynamical correlation energy and the completeactive space energy, respectively.
By utilizing the twopoint extrapolation protocol proposed by Karton and Martin [34], the CAS energies are extrapolated to CBS limit.
(2) 
where E_{∞}^{CAS} is the energy when X → ∞ and a=5.34 is an effective decay exponent.
The USTE technique [2426] has been successfully performed to extrapolate the dc energies in MRCI (Q) calculations, which takes the following form:
(3) 
(4) 
where A_{5}(0) =0.0037685459, c=1.17847713 and α=3/8 are the universaltype parameters [25]. Thus, Eq.(3) is then transformed into an (E_{∞}, A_{3}) twoparameter rule, which is actually used for the practical procedure of extrapolation [30,33,35]. Thus, the dc energies were extrapolated to the CBS limit by utilizing USTE extrapolation scheme.
C. APEFs of LiS^{n} (n=1, 0, 1)The APEFs of LiS^{n} (n=1, 0, 1) are written as the formalism developed by Aguado and Paniagua [27,28], which is expressed as a sum of two terms corresponding to the shortrange and longrange potentials,
(5) 
(6) 
where the diatomic potentials which tend to zero as R_{LiS}→∞. The shortrange potentials which tend to infinite value when R_{LiS}→0 takes the following expression
(7) 
The parameters in Eqs.(6) and (7) are obtained by fitting the ab initio energies calculated using AVXdZ (X=T, Q), which are then extrapolated to the CBS limit. Moreover, both the CV and DK are also considered, which are added to the CBS results and then employed to model the APEFs, here and after denoted as CBS+CV+DK APEFs. The nonlinear parameters β_{i} (i=1, 2) and linear parameters a_{i} (i=0, 1, 2, …, n) in Eq.(6) and Eq.(7) are gathered in Table Ⅰ.
Ⅲ. RESULTS AND DISCUSSION A. The PECsThe results of LiS^{−}(^{1}Σ^{+}), LiS (^{2}∏) and LiS^{+}(^{3}Σ^{−}) PECs calculated at CBS+CV+DK level are shown in Fig. 1. It shows that the CBS+CV+DK PEC of LiS^{−}(^{1}Σ^{+}) is deeper than the CBS one. While, it can be seen through the other two figures that the CBS PECs of LiS (^{2}∏) and LiS^{+}(^{3}Σ^{−}) are both deeper than CBS+CV+DK PECs. For comparison, both the fitted MRCI (Q)/CBS+CV+DK APEFs and the ab initio energies are displayed in Fig. 2. Shown in this figure are also the difference between APEFs and the ab initio energies. As can be seen from this figure, the modeled APEFs accurately mimic the ab initio energies. To evaluate the fitting quality of the fitted APEFs, we calculated the rootmean square derivation (RMSD) using the following equation:
(8) 
where N is the number of points utilized in the fitting process, V_{fit} are the energies obtained from the fitted APEFs and V_{ab} are MRCI (Q)/CBS+CV+DK energies, respectively. The values of △E_{RMSD} are 0.1086, 0.1423, and 0.1335 kcal/mol for LiS^{−}(^{1}Σ^{+}), LiS (^{2}∏) and LiS^{+}(^{3}Σ^{−}) respectively, showing high accuracy of the fitted APEFs.
B. Spectroscopic constantsBy utilizing APEFs of LiS^{−}(^{1}Σ^{+}), LiS (^{2}∏) and LiS^{+}(^{3}Σ^{−}), the spectroscopic constants are calculated, which are tabulated in Table Ⅱ. The other theoretical results [4,6,13,16] are also tabulated in Table Ⅱ for convenient comparison. It can be seen from Table Ⅱ that the values of R_{e}, D_{e}, ω_{e}, ω_{e}χ_{e}, B_{e}, and α_{e} of LiS (^{2}∏) together with the other experiment [5] and theoretical data [4,6,16] for convenient comparison. The R_{e} of LiS (^{2}∏) decreases as the basis set increases from AVTdZ to CBS+CV+DK APEF. The values of D_{e} calculated from CBS+CV+DK APEF is 3.2776 eV, which differs from the experimental [5] value by 0.022 eV. Comparing the results calculated from CBS+CV+DK APEF with those of experimental [5] and theoretical data [6], the deviation of ω_{e} and B_{e} are 0.39% and 1.14%, 0.14% and 0.46%, respectively. The values of ω_{e}χ_{e} and α_{e} calculated from CBS+CV+DK APEF differ from those of CBS APEF, by 4.3% and 2.8%, respectively.
For LiS^{+}(^{3}Σ^{−}), the equilibrium R_{e} obtained from the AVTdZ, AVQdZ, CBS and CBS+CV+DK APEFs are 2.5147, 2.4883, 2.4874, and 2.4857 Å, respectively. As can be seen from this table that the equilibrium bond lengths predicted from the CBS and CBS+CV+DK APEFs are only 0.0414 and 0.0397 Å larger than the theoretical data in Ref.[6]. The dissociation energy (D_{e}) calculated from CBS+CV+DK APES is 0.7625 eV, which is only 0.0191 and 0.0685 eV smaller than the CBS value and theoretical data [13]. The vibration frequency is calculated to be 326.984 cm^{1} at the CBS+CV+DK level, which differs from the theoretical values in Refs.[6] and [13] only by 26.016 and 49.016 cm^{1}, respectively. Comparing the results from the present CBS+CV+DK APEF with CBS APEF, the deviations of ω_{e}, B_{e}, α_{e}, and ω_{e}χ_{e} are 3.268%, 0.143%, 0.521% and 3.934%, respectively.
For the LiS^{−}(^{1}Σ^{+}), Table Ⅱ, D_{e} increases monotonically from AVTdZ to CBS APEFs, and the deepest well depth is obtained from CBS+CV+DK APEF, with the difference of 0.0668 and 0.3222 eV from those of the CBS APEF and theoretical results [16]. R_{e} decreases and B_{e} increases from the result of AVTdZ to CBS+CV+DK APEFs. The differences of vibrational frequeny ω_{e} obtained from CBS+CV+DK APEFs are 0.051 and 4.738 cm^{1}, compared with CBS and theoretical results [16], respectively. Comparing the results from the present CBS APEF with the those of CBS+CV+DK APEF, the deviation of R_{e}, ω_{e}, ω_{e}χ_{e}, B_{e}, α_{e} are 0.283%, 0.0078%, 2.4%, 0.57% and 1.29%, respectively.
C. Vibrational energy levelsBy solving the radical Schrödinger equation of the nuclear motion with LEVEL 7.5 program [9], the vibrational energy levels are calculated. The radical Schrödinger equation is written as
(9) 
where E_{ν, J} is eigenvalues, ψ_{υ, J} is eigenfunction, V(r) is the potential energy, J and υ are the rotational and vibrational quantum number, r and μ are the internuclear distance and the reduced mass of the molecule, respectively. For a given vibrational level, the rotational sublevels can written as
(10) 
where G(υ) is the vibrational level, B_{υ} is inertial rotation constant, and D_{ν}, H_{ν}, L_{ν}, M_{ν}, N_{ν}, and O_{υ} are the centrifugal distortion constants, respectively.
The complete set of vibrational states for of LiS^{n} (n=1, 0, 1) are calculated when J=0, by solving Eq.(10) numerically. Table ⅢⅣ gather the classical turning points (R_{min}, R_{max}), the inertial rotation constants B_{ν} and the vibrational levels G(υ) calculated from the CBS+CV+DK APEFs for LiS, LiS^{} and LiS^{+} respectively. Here, due to the length limitation, we only tabulate the result of 21 vibrational states. In Table Ⅲ, we present the corresponding results of LiS (^{2}∏). It can be found that vibrational levels G(υ) of CBS+CV+DK APEF show small difference from the CBS APEF, the deviations of v=0, 1, 2, 3, and 4 are 0.21%, 0.065%, 0.021%, 0.067% and 0.085%, respectively. It can be seen from Table Ⅳ, the differences between the CBS and CBS+CV+DK APEFs of LiS^{+}(^{3}Σ^{−}) are only 3.69%, 3.1%, 3.06%, 4.1% and 2.88% for v=0, 1, 2, 3, and 4, respectively. Table Ⅴ demonstrates the vibrational levels G(υ) of LiS^{−}(^{1}Σ^{+}) from CBS APEF and CBS APEF, showing the deviations for v=0, 1, 2, 3, and 4 are 0.023%, 0.040%, 0.060%, 0.080% and 0.10%, respectively. Whereas, according to the high quality of CBS+CV+DK APEF, the results are accurate and reliable. As a result, the present work provides more accurate and complete investigation on the LiS^{n} (n=1, 0, 1) system.
Ⅳ. CONCLUSIONThe PECs, spectroscopic constants, classical turning points and vibrational levels are studied for LiS, LiS^{} and LiS^{+} systems. The ab initio energies are calculated at the MRCI/AVXdZ (X=T, Q) levels of the theory which are then extrapolated to the CBS limit. The the relativistic effect and corevalence correlation are also considered. Excellent agreement on spectroscopic parameters is obtained between the present result and other theoretical and experimental results. It can be concluded that the present work provide more accurate and complete investigations on the spectroscopic constants and vibrational manifolds of LiS (^{2}∏), LiS^{−}(^{1}Σ^{+}) and LiS^{+}(^{3}Σ^{−}), respectively.
Ⅴ. AcknowledgmentsThis work was supported by the National Natural Science Foundation of China (No.11304185), Taishan scholar project of Shandong Province, China Postdoctoral Science Foundation (No.2014M561957), and Postdoctoral Innovation Project of Shandong Province (No.201402013), Shandong Provincial Natural Science Foundation (No.ZR2014AM022). The authors gratefully acknowledge Dr. S. Li for useful discussion in this work.
[1]  D. E. Jensen, and G. A. Jones, Combust. Flame 41 , 71 (1981). DOI:10.1016/00102180(81)900407 
[2]  B. Gustafsson, Ann. Rev. Astron. Astrophys. 27 , 701 (1989). DOI:10.1146/annurev.aa.27.090189.003413 
[3]  Y. Q. Xu, W. C. Peng, and Y. Q. Cai, Chin. J. Chem. Phys. 33 , 749 (2016). 
[4]  H. Partridge, S. R. Langho, and C.W. Bauschlicher Jr., J. Chem. Phys. 88 , 6431 (1988). DOI:10.1063/1.454429 
[5]  M.A. Brewster, and M. A. Ziurys, Chem. Phys. Lett. 349 , 249 (2001). DOI:10.1016/S00092614(01)012027 
[6]  E.P.F. Lee, and T. G. Wright, Chem. Phys. Lett. 397 , 194 (2004). DOI:10.1016/j.cplett.2004.08.104 
[7]  T. Helgaker, W. Klopper, H. Koch, and J. Noga, J. Chem. Phys. 106 , 9639 (1997). DOI:10.1063/1.473863 
[8]  A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen, and A. K. Wilson, Chem. Phys. Lett. 286 , 243 (1998). DOI:10.1016/S00092614(98)001110 
[9]  R. J. Le Roy, LEVEL 7. 5: A Computer Program for Solving the radial Schrödinger Equation for Bound and Quasibound Levels, University of Waterloo Chemical Physics Report CP655, (2002). 
[10]  P.J. Knowles, and H.J. Werner, Chem. Phys. Lett. 115 , 259 (1985). DOI:10.1016/00092614(85)800257 
[11]  H.J. Werner, and P.J. Knowles, J. Chem. Phys. 89 , 5803 (1988). DOI:10.1063/1.455556 
[12]  P.J. Knowles, and H.J. Werner, Chem. Phys. Lett. 145 , 514 (1988). DOI:10.1016/00092614(88)874128 
[13]  F. Khadri, H. Ndome, S. Lahmar, Z. B. Lakhdar, and M. Hochlaf, J. Mol. Spectro. 237 , 232 (2006). DOI:10.1016/j.jms.2006.04.001 
[14]  T.H. Dunning Jr., J. Chem. Phys. 90 , 1007 (1989). DOI:10.1063/1.456153 
[15]  D.E. Woon, and T.H. Dunning Jr., J. Chem. Phys. 98 , 1358 (1993). DOI:10.1063/1.464303 
[16]  A. I. Boldyrev, J. Simons, and P.V. R. Schleyer, J. Chem. Phys. 99 , 8793 (1993). DOI:10.1063/1.465600 
[17]  P.C. Hariharan, and J.A. Pople, Theor. Chim. Acta 28 , 213 (1973). DOI:10.1007/BF00533485 
[18]  M. J. Frisch, J. A. Pople, and J. S. Binkley, J. Chem. Phys. 80 , 3265 (1984). DOI:10.1063/1.447079 
[19]  T. Clark, J. Chandrasekhar, G. W. Spitznagel, and P.V. R. Schleyer, J. Comput. Chem. 4 , 294 (1983). DOI:10.1002/(ISSN)1096987X 
[20]  A.D. McLean, and G. S. Chandler, J. Chem. Phys. 72 , 5639 (1980). DOI:10.1063/1.438980 
[21]  R. Krishnan, and J. A. Pople, Int. J. Quantum Chem. 14 , 91 (1978). DOI:10.1002/(ISSN)1097461X 
[22]  S. Olivella, J. M. Anglada, Solé A., and J.M. Boll, Chem. A Eur. J. 10 , 3404 (2004). DOI:10.1002/(ISSN)15213765 
[23]  H.J. Werner, and P. J. Knowles, J. Chem. Phys. 89 , 5803 (1988). DOI:10.1063/1.455556 
[24]  A.J. C. Varandas, J. Chem. Phys. 126 , 244105 (2007). DOI:10.1063/1.2741259 
[25]  A.J. C. Varandas, J. Chem. Phys. 127 , 114316 (2007). DOI:10.1063/1.2768356 
[26]  A.J. C. Varandas, J. Chem. Phys. 113 , 8880 (2000). DOI:10.1063/1.1319644 
[27]  A. Aguado, and M. Paniagua, J. Chem. Phys. 96 , 1265 (1992). DOI:10.1063/1.462163 
[28]  A. Aguado, C. Tablero, and M. Paniagua, Comput. Phys. Commun. 108 , 259 (1998). DOI:10.1016/S00104655(97)001355 
[29]  H. J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, P. Celani, W. Györffy, D. Kats, T. Korona, R. Lindh, A. Mitrushenkov, G. Rauhut, K. R. Shamasundar, T. B. Adler, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, E. Goll, C. Hampel, A. Hesselmann, G. Hetzer, T. Hrenar, G. Jansen, C. Köppl, Y. Liu, A. W. Lloyd, R. A. Mata, A. J. May, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklaβ, D. P. O'Neill, P. Palmieri, D. Peng, K. P uger, R. Pitzer, M. Reiher, T. Shiozaki, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, and M. Wang, MOLPRO Version 2012. 1, (2012). (www.molpro.net) 
[30]  L. L. Zhang, S. B. Gao, Q. T. Meng, and Y. Z. Song, Chin. Phys B24 , 013101 (2015). 
[31]  Y. L. Liu, H. S. Zhai, X. M. Zhang, and Y. F. Liu, Chem. Phys. 425 , 156 (2013). DOI:10.1016/j.chemphys.2013.09.002 
[32]  S.Y. Liu, and H.S. Zhai, At. Mol. Sci 6 , 197 (2015). 
[33]  Y.Z. Song, and A J C. Varandas, J. Chem. Phys. 130 , 134317 (2009). DOI:10.1063/1.3103268 
[34]  A. Karton, and J. M. L. Martin, Theor. Chem. Acc. 115 , 330 (2006). DOI:10.1007/s0021400500286 
[35]  L. L. Zhang, J. Zhang, Q. T. Meng, and Y. Z. Song, Phys. Script. 90 , 035403 (2015). DOI:10.1088/00318949/90/3/035403 