Chinese Journal of Chemical Physics  2017, Vol. 30 Issue (2): 128-134

The article information

En Cao, Shuang Liu, Yu-zhi 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
Chinese Journal of Chemical Physics, 2017, 30(2): 128-134
化学物理学报, 2017, 30(2): 128-134

Article history

Received on: November 17, 2016
Accepted on: January 20, 2017
Accurate Theoretical Study of LiS Radical and Its Singly Charged Cation and Anion in their Ground Electronic State
En Cao, Shuang Liu, Yu-zhi Song     
Dated: Received on November 17, 2016; Accepted on January 20, 2017
School of Physics and Electronic, Shandong Normal University, Jinan 250014, China
*Author to whom correspondence should be addressed. Yu-zhi Song,
Abstract: Potential energies of LiS (2Π), LiS-(1Σ+) and LiS+(3Σ-) are calculated by using the multireference configuration interaction method including Davidson correction and the augmented correlation-consistent basis sets aug-cc-PV (X+d) Z (X=T, Q). Such obtained potential energies are subsequently extrapolated to the complete basis set limit. Both the core-valence correction and the relativistic effect are also considered. The analytical potential energy functions are then obtained by fitting such accurate energies utilizing a least-squares fitting procedure. By using such analytical potential energy functions, we obtain the accurate spectroscopic parameters, complete set of vibrational levels and classical turning points. The present results are compared well with the experimental and other theoretical work.
Key words: Analytical potential energy function    Spectroscopic constants    Vibrational energy levels    

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/sub-millimeter spectrum, Brewster and Ziurys [5] obtained spectroscopic constants and equilibrium bond length in the ground state of LiS (X2∏). 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 two-point 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 self-consistent field (CASSCF) method [10] which was then followed by the internally contracted multi-reference configuration interaction approach (MRCI) [11,12], Khadri et al. [13] carried out detailed studies of LiS+ with the correlation-consistent basis sets of Dunning, cc-PV5Z [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(1Σ+) with polarized spilt-valence basis sets (6-311+G*) [17-20] at correlated second-order (MP2) levels and self-consistent field method (SCF). The equilibrium geometries were used to evaluate electron correlation in the frozen-core approximation by full fourth order [21] moller-plesset perturbation theory and the (U) Q-CISD (T) method [22] for LiMn (n=+1, 0, -1; M=Li, Be, P, C, N, O, F).

In order to obtain the accurate PECs of LiSn (n=-1, 0, +1), by employing the MRCI (Q) method [23], we carried out detailed studies of LiSn (n=-1, 0, +1) systems with the standard ugmented correlation-consistent basis sets, aug-cc-PV (T+d) Z (AVTdZ) and aug-cc-PV (Q+d) Z (AVQdZ). The core-valence (CV) correlation is carried out with the CV basis set aug-cc-PCVQZ, and the relativistic correction is also taken into account by using the aug-cc-PVQZ-DK basis set. In particular, the uniform singlet-pair and triplet-pair extrapolation (USTE) protocol [24-26] 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 calculations

The ab initio calculations are carried out by using MOLPRO 2012 program [29]. In order to obtain the high-level PECs of LiSn (n=-1, 0, 1), the potential energies are calculated for the internuclear separation ranging from 0.5 a0 to 35 a0 with the interval of 0.05 a0, which declines to 0.01 a0 in the vicinity of the equilibrium geometries. All calculations are carried out at the MRCI (Q) level using the CASSCF wave function [10-12] as the reference, which has been applied to many diatomic molecules [30-32]. In the ab initio calculations of PECs of LiSn (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 high-expenent core-polarization d functions as performed for the second row atoms. Both the core-valence correction and the relativistic effect are considered. The core-valence (CV) correction is taken into account with the CV basis set aug-cc-PCVQZ, and the relativistic correction is carried out with the aug-cc-PVQZ-DK basis set, respectively. We employ C2v point group symmetry in the ab initio calculations, which includes four irreducible representations, namely A1, B1, B2 and A2, respectively. For LiSn (n=-1, 0, 1), 8 orbitals (4a1+2b1+2b2) are confirmed as the active space. Thus, in order to select molecular state, we choose the A1 irreducible representation of the C2v point group and carry out the two-state average calculation.

B. Extrapolation to CBS limit

The MRCI (Q) electronic energy can be treated in spilt form, which can be written as [33]


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 complete-active space energy, respectively.

By utilizing the two-point extrapolation protocol proposed by Karton and Martin [34], the CAS energies are extrapolated to CBS limit.


where ECAS is the energy when X → ∞ and a=5.34 is an effective decay exponent.

The USTE technique [24-26] has been successfully performed to extrapolate the dc energies in MRCI (Q) calculations, which takes the following form:


where A5(0) =0.0037685459, c=-1.17847713 and α=-3/8 are the universal-type parameters [25]. Thus, Eq.(3) is then transformed into an (E, A3) two-parameter 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 LiSn (n=-1, 0, 1)

The APEFs of LiSn (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 short-range and long-range potentials,


where the diatomic potentials which tend to zero as RLiS→∞. The short-range potentials which tend to infinite value when RLiS→0 takes the following expression


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 βi (i=1, 2) and linear parameters ai (i=0, 1, 2, …, n) in Eq.(6) and Eq.(7) are gathered in Table Ⅰ.

Table Ⅰ Fitted parameters of LiS(1Σ+), LiS (2Π) and LiS+(3Σ) APEFs in Eqs.(6) and (7).

The results of LiS(1Σ+), LiS (2∏) and LiS+(3Σ) PECs calculated at CBS+CV+DK level are shown in Fig. 1. It shows that the CBS+CV+DK PEC of LiS(1Σ+) 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+(3Σ) 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 root-mean square derivation (RMSD) using the following equation:

FIG. 1 PECs of LiS+(3Σ), LiS (2Π), and LiS(1Σ+). The PECs of LiS (2Π) obtained from the CBS APEF and the CBS+CV+DK APEFs.
FIG. 2 APEFs of LiS+(3Σ), LiS (2Π), and LiS(1Σ+). The circles indicate the MRCI (Q)/CBS+CV+DK energies, while lines are from the fitted the APEFs.

where N is the number of points utilized in the fitting process, Vfit are the energies obtained from the fitted APEFs and Vab are MRCI (Q)/CBS+CV+DK energies, respectively. The values of △ERMSD are 0.1086, 0.1423, and 0.1335 kcal/mol for LiS(1Σ+), LiS (2∏) and LiS+(3Σ) respectively, showing high accuracy of the fitted APEFs.

B. Spectroscopic constants

By utilizing APEFs of LiS(1Σ+), LiS (2∏) and LiS+(3Σ), 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 Re, De, ωe, ωeχe, Be, and αe of LiS (2∏) together with the other experiment [5] and theoretical data [4,6,16] for convenient comparison. The Re of LiS (2∏) decreases as the basis set increases from AVTdZ to CBS+CV+DK APEF. The values of De 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 ωe and Be are 0.39% and 1.14%, 0.14% and 0.46%, respectively. The values of ωeχe and αe calculated from CBS+CV+DK APEF differ from those of CBS APEF, by 4.3% and 2.8%, respectively.

Table Ⅱ Spectroscopic constants of LiS (2Π), LiS+(3Σ), and LiS(1Σ+).

For LiS+(3Σ), the equilibrium Re 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 (De) 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 ωe, Be, αe, and ωeχe are 3.268%, 0.143%, 0.521% and 3.934%, respectively.

For the LiS(1Σ+), Table Ⅱ, De 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]. Re decreases and Be increases from the result of AVTdZ to CBS+CV+DK APEFs. The differences of vibrational frequeny ωe 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 Re, ωe, ωeχe, Be, αe are 0.283%, 0.0078%, 2.4%, 0.57% and 1.29%, respectively.

C. Vibrational energy levels

By 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


where Eν, J is eigenvalues, ψυ, 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


where G(υ) is the vibrational level, Bυ is inertial rotation constant, and Dν, Hν, Lν, Mν, Nν, and Oυ are the centrifugal distortion constants, respectively.

The complete set of vibrational states for of LiSn (n=-1, 0, 1) are calculated when J=0, by solving Eq.(10) numerically. Table Ⅲ- gather the classical turning points (Rmin, Rmax), the inertial rotation constants Bν 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+(3Σ) 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(1Σ+) 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 LiSn (n=-1, 0, 1) system.

Table 3 The vibrational levels G(ν), classical turning points R, and rotational constant Bν of the first 21 vibrational states for LiS (2Π) when J=0 predicted by the CBS+CV+DK APEF and CBS APEF.
Table 4 The vibrational levels G(ν), classical turning points, and rotational constant Bν of the first 21 vibrational states for LiS+(3Σ) when J=0 predicted by the CBS+CV+DK APEF.
Table Ⅴ The vibrational levels, classical turning points, and rotational constant of the first 21 vibrational states for LiS(1Σ+) when J=0 predicted by the CBS+CV+DK APEF and CBS APEF.

The 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 core-valence 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(1Σ+) and LiS+(3Σ), respectively.

Ⅴ. Acknowledgments

This 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 Post-doctoral 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.

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曹恩, 刘爽, 宋玉志     
山东师范大学物理与电子科学学院, 济南 250014
摘要: 利用多组态相互作用方法和aug-cc-PV (X+d) Z (X=T, Q) 基组计算了LiS (2Π)、LiS-(1Σ+) 和LiS+(3Σ-) 的势能, 并考虑了零点能修正、核价相关和相对论效应.利用基组外推方法把能量外推到完备基组极限, 得到了更加精确的能量.利用最小二乘法拟合这些能量点获得了解析的势能函数, 计算得到了精确的光谱常数、振动能级和经典拐点.计算结果与实验和其它理论值符合得很好.
关键词: 解析势能函数    光谱常数    振动能级