Spectroscopic constants and anharmonic force field of dithioformic acid and its isomers: a theoretical study

The potential astronomical interest dithioformic acid (trans-HC(= S)SH) exists five isomers and has received considerable attention of astronomical observation in recent years. The different positions of H atoms of five isomers lead to diverse point groups, dipole moments, and spectroscopic constants. The anharmonic force field and spectroscopic constants of them are calculated using CCSD(T) and B3LYP employing correlation consistent basis sets. Molecular structures, dipole moments, rotational constants, and fundamental frequencies of trans-HC(= S)SH are compared with the available experimental data. The B3LYP/Gen = 5 and CCSD(T)/Gen = Q results can reproduce them well. Molecular structures, dipole moments, relative energies, spectroscopic constants of cis-HC(= S)SH, and dithiohydroxy carbene (DTHC) are also calculated. The new data obtained in this study are expected to guide the future high resolution experimental work and to assist astronomical search for CH2S2.


Introduction
Sulfur is a highly abundant element in the galaxy, and more than 20 sulfur compounds have been detected in some astrophysical environments [1,2]. Especially, some C x H y S z isomers have received considerable attention of astronomical observation in recent years. So far, some sulfur-bearing molecules [2][3][4][5][6][7] have been identified in the interstellar medium (ISM). In order to systematically understand the sulfur chemistry in the astrophysical environment, new molecules with multiple sulfur atoms are required to search in some planetary atmospheres or surfaces. The astronomical search of one molecule can be achieved directly by the pure rotational spectroscopy [8]; thus, the potential astronomical interest molecules, CH 2 XY (X, Y = O, S) isomers (such as HC(= O)SH, HC(= S)OH, and HC(= S)SH), have received experimental or theoretical investigations of its physical properties and rotational transitions [9,10]. Although none of the CH 2 S 2 isomers has been detected in the space, they are also expected to exist in L183 or Barnard 1, which are both S-rich prestellar cores [10].
There are five isomers for the simplest dithio carboxylic homolog CH 2 S 2 : dithioformic acid (DTFA) exhibits transand cis-HC(= S)SH conformations and dithiohydroxy carbene (DTHC) contains 2tt-, 2 cc-, and 2tc-CH 2 S 2 conformers (collected in Fig. 1). These isomers have same backbone of S-C-S and different position of H atoms. The shift of intramolecular hydrogen makes them have different total energies and dipole moments. The energy difference of CH 2 S 2 isomers can be used as a "thermometer" in the astrophysical environments, which is extremely cold conditions and usually too harsh to reproduce it on the earth [11]. On the other hand, the various styles of molecular dipole moment lead to the diverse rotational spectrum. Thus, the geometry structures, total energies, dipole moments, and vibrational and rotational spectra of CH 2 S 2 isomers were investigated [9,10,[12][13][14][15][16][17][18][19][20][21][22][23][24].
As far as the authors know, there are some experimental papers which focused on vibrational or rotational spectra for CH 2 S 2 isomers [10,18,[22][23][24]. In 1971, the IR spectrum of trans-HC(= S)SH was firstly reported in a short communication [24], and several vibrational wave numbers were determined without band assignment. In 1989, the low-resolution IR spectrum of DTFA was observed at the range of 475-3100 cm −1 [18]. Six and four fundamental frequencies were assigned for trans-and cis-HC(= S)SH, respectively. In 1978, the rotational transitions of ground state of DTFA were firstly identified using microwave spectrum in the region of 18-40 GHz by Bak et al. [23]. The original rotational spectroscopy parameters of DTFA were determined, such as ground-state rotational constants ( A 0 , B 0 , and C 0 ), two quartic centrifugal distortion constants ( Δ J and Δ JK ), asymmetry parameter K , and inertial defects Δ . The "iterative" structure of trans-HC(= S)SH isomer was reported using Kraitchman technique. Additionally, the dipole moments and a of DTFA were derived using rigid model. However, only a-type transitions were measured and the estimated uncertainties of rotational constants A 0 were 86 and 380 MHz for trans-and cis-HC(= S)SH, respectively. One year later, Bak and his coworkers reinvestigated the microwave spectrum of DTFA in the region of 18.6-40 GHz [22]. The same parameters were obtained for trans-HC(= 32 S) 32 SH, HC(= 34 S) 32 SH, HC(= 32 S) 34 SH, HC(= 32 S) 32 SD, and cis-HC(= 32 S) 32 SD. In order to improve the accuracy of rotational spectroscopic constants of DTFA, more than 300 lines of rotational transitions were measured by Prudenzano et al. [10]. The ground-state rotational constants, complete quartic, and four sextic centrifugal distortion constants were obtained accurately in the range of the millimeter and sub-millimeter. With the development of quantum-chemical calculations, it is possible to obtain the spectroscopic constants of astrochemical molecules by quantum-chemical calculations. And the computational spectroscopy is important to investigate astrochemistry as the molecular complexity in space [11]. In 1979, the optimized geometries, relative energies, and dipole moment components of DTFA were calculated using LCAO-MO-SCF theory [22]. The theoretical calculations of molecular structure parameters of DTFA have been reported [9,10,[13][14][15][16][17][18]. The main methods used by them were SCF, SCF-MO, and CCSD(T); and the main basis sets were 3-21G*, 6-31G*, 6-31G**, and CBS + CV. For the vibrational spectra of DTFA isomers, only harmonic wave numbers were calculated using SCF/6-31G* [18], SCF-MO/6-31G* [19], and STO-3G/DZP [16] levels of theory. The existence of several isomers implies the potential large amplitude motions (LAMs) in molecules [11]. The accurate prediction of the vibrational spectra might therefore be required using high-level quantum-chemical theory for these isomers. For the rotational spectra of DTFA, the three equilibrium rotational constants A e , B e , and C e , were predicted using SCF/6-31G* level of theory [18]. In order to obtain rotational constants and to predict quartic and sextic centrifugal distortion constants theoretically, the high level of theory is needed for DTFA. Only structural parameters and relative energies of the three isomers of DTHC have been theoretically studied [13,17]. The relative energies of the lowest-excited vibrational states for the most stable isomer are similar to those of the vibrational ground states for less stable ones [11]. This may be a reason to be difficult to observe the rotational or vibrational spectra of the three DTHC isomers.
These quantum-chemical calculations of CH 2 S 2 mentioned above have paralleled the experimental measurements and only molecular geometry parameters of DTFA were determined employing CCSD(T) method. All the previous studies show that some key spectral parameters of the five isomers of CH 2 S 2 are still absent. Therefore, the purpose of this paper is to obtain the vibrational and rotational spectra with quantum-chemical computations for trans-HC(= S)SH, cis-HC(= S)SH, 2tt-CH 2 S 2 , 2 cc-CH 2 S 2 , and 2tc-CH 2 S 2 .
The mixed basis set Gen = Q presented below is denoted that cc-pVQZ basis set is employed for the carbon and hydrogen atoms and the d-augmented basis set cc-pV(Q + d)Z for sulfur atoms. Similarly with the denotation of Gen = Q, the mixed basis set Gen = 5 is denoted that cc-pV5Z basis set is employed for the light atoms (H and C), and cc-pV(5 + d)Z basis set for sulfur atoms. Cubic and semi-diagonal quartic force constants of the five CH 2 S 2 isomers in normal coordinate are calculated using B3LYP method. No orbital has been kept frozen during the B3LYP calculations.
CCSD(T) (coupled-cluster singles and doubles with perturbative triples) is performed with the help of CFOUR [31] program. The CCSD(T) calculations employ correlation consistent basis sets [28][29][30], cc-pVQZ, and the abovementioned Gen = Q mixed basis set. The cc-pVQZ basis set is abbreviated as VQZ in the following sections. Cubic force constants of the five CH 2 S 2 isomers in normal coordinate are calculated using CCSD(T) method. The frozen-core approximation is employed in CCSD(T) method.
Within the constraint of C s (DTFA and 2tc-CH 2 S 2 ) and C 2v (2tt-and 2 cc-CH 2 S 2 ) point groups, the geometries of the five CH 2 S 2 isomers are optimized with analytic gradients. Based on the optimized geometries, harmonic force fields of these isomers are evaluated analytically. The normal modes and the harmonic spectroscopic constants are computed in the usual manner [32][33][34][35].
Recently, the semi-empirical procedure named "best estimate" [36] starts from various contributions with the computational level as high as possible, and then combined them together. The investigations show that the effective approach can obtain accurate molecular structures (with an accuracy of 0.001-0.002 Å for bond distances, and 0.05-0.1° for bond angles, respectively) with this composite scheme [11,37,38]. As an attempt, empirical procedure is partly used for B3LYP formulas to evaluate molecular structure of trans-HC(= S)SH.
For the molecular structure, complete basis set (CBS) limit is derived based on the similar convergence behavior of structural parameters and energy. Considering expression (7) of Ref. [39], CBS limit of molecular parameters is carried out by the following extrapolation form where r(n) is the B3LYP/cc-pVnZ structural parameter. And the n = 4 and 5 values are used in expression (4).
The effect of diffuse functions (Aug) is included using the following difference: where r(AVQZ) and r(VQZ) are molecular geometries employed by aug-cc-pVQZ and cc-pVQZ basis sets, respectively, deriving from B3LYP method. Analogously, the effect of tight-d augmented valence is used for the atom of sulfur: Completely, the empirical best theoretical estimates (simply denoted as "Best") of molecular structure are provided by the following expression (2) Δr(Aug) = r(AVQZ) − r(VQZ) In later sections, "Best 1" and "Best 2" values respectively refer to n = 4 and 5 results of r(Best).

Results and discussion
The spectroscopic constants and geometry structures for CH 2 S 2 isomers are given in Tables 1, 2, 3, 4, 5, 6, 7 and 8. They are compared with the corresponding experimental [10,18,[22][23][24] or theoretical data [10,13,[16][17][18][19] whenever available. Figures 2 and 4 are given by the virtual multi-frequency spectrometer, VMS [40,41]. Table 1 Molecular equilibrium geometry structures and dipole moments of CH 2 S 2 isomers (Bond lengths in Å, bond angles in deg, dipole moments in Debye, and total energies in Hartree). In this work, the parameters are determined by B3LYP and CCSD(T) methods employing Gen = 5 and Gen = Q, respectively. The correlative data of "Best 1" and "Best 2" are given in Table S1 a The molecular structure derived based on the ground-state rotational constants by an iterative procedure in Ref [22]. The value of parameter r(C1-H4) was assumed fixed in the Ref [22] b The calculated results of DTFA were obtained by CCSD(T)/CBS + CV level of theory with the CFOUR program c MP2/3-21G* level of theory was employed using Gaussian82 program d The MP2/6-31G(d,p) and B3LYP/6-311 + + G** levels of theory were employed for the determination of geometries and total energy, respectively

Geometry structures, dipole moments, and energies
The ab initio structures of five CH 2 S 2 isomers are calculated at B3LYP and CCSD(T) methods using mixed basis sets Gen = 5 and Q, respectively. The computed equilibrium structures, experimentally derived results (Exp in short) [22], and previously calculated values (Pre in short) [10,13,17] are shown in Table 1. Figure 1 illustrates the molecular structures, and all the five isomers have a planar form, because only zero torsional angle has the minima torsional potential [13,18].
The bond length of other three isomers vary little with B3LYP/Gen = 5 and CCSD(T)/Gen = Q levels of theory except for the 2tt-and 2 cc-CH 2 S 2 isomers. Precisely, it can be found that the biggest discrepancy of C-S bond in 2 cc-and 2tt-CH 2 S 2 between B3LYP/Gen = 5 and CCSD(T)/ Gen = Q is about 0.014 and 0.012 Å, respectively. It is about 3° for S-C-S bond angle in the 2 cc-and 2tt-CH 2 S 2 isomers, which needs more theoretical or experimental investigations on 2 cc-and 2tt-CH 2 S 2 in the future. Hence, the B3LYP/Gen = 5 level of theory can give alternative results for DTFA. By comparing our calculated equilibrium geometries to the experimental results [22] of trans-HC(= S)SH, a good agreement can be found: the most calculated results agree within the uncertainties.
As the "best estimate" can represent alternative results for sulfur containing molecules, the empirical B3LYP values (Best 1 and Best 2) of trans-HC(= S)SH are also given in Table 1. And Table S1 lists the data used to carry out the best estimate values. For the C-S (C-H) bond, the "Best" Table 2 Equilibrium and ground states rotational constants (in MHz), asymmetry parameters, and inertial defects (in amu*Å. 2 ) for trans-HC(= S)SH a The inertial defect was calculated using conversion factor 505,376 MHz amu Å. 2 and the asymmetry parameter was also calculated by the definition b The values were obtained by the microwave spectrum in the range of 18-40 GHz c The data were calculated using SCF/6-31G* B3LYP CCSD(T) Exp [10]a Exp [22,23]b Pre [18] Table 3 Equilibrium and ground states rotational constants (in MHz), asymmetry parameters, and inertial defects (in amu Å. 2 ) for CH 2 S 2 isomers. In this work, the parameters are determined by B3LYP and CCSD(T) methods employing Gen = 5 and Gen = Q, respectively a The inertial defect was calculated using conversion factor 505,376 MHz amu Å. 2 and the asymmetry parameter was also calculated by the definition b The values were obtained by the microwave spectrum in the range of 18-40 GHz c The data were calculated using SCF/6-31G* values are about 0.005 Å (0.001 Å) shorter than CCSD(T)/ Gen = Q, which is often considered "gold standard" for geometry optimizations [38]. It is also worth noting that Best 1 (employing VTZ and VQZ) has significant basis set truncation effect and provides relatively reliable bond lengths. While the "Best" results of bond angles (S-C-S and C-S-H) are not satisfactory, maybe it needs another suitable corrections, such as core-valence correction described in Ref. [38].
For the five isomers, the bond angles are influenced obviously by the position of H atoms, while the bond lengths are not sensitive to it. The values of C-H and S-H bond length are similar, which are about 1.08 and 1.34 Å, respectively.
Relatively, the bond angles lie in a wide range. Especially, the discrepancies of S-C-S bond angle at B3LYP/Gen = 5 level of theory are about 5° and 13° for DTFA and DTHC, respectively. The opening of heavy-atom backbone S-C-S and giving way to H atoms [19] lead to larger S-C-S bond angle in trans-HC(= S)SH (2 cc-CH 2 S 2 ) than in cis-HC(= S) SH (2tt-and 2tc-CH 2 S 2 ). And it is suggested that the cis-HC(= S)SH, 2tt-, and 2tc-CH 2 S 2 isomers can be formed by adding a hydrogen atom to the trans-HSCS radical [16]. Similarly, adding a hydrogen atom to the cis-HSCS radical can form trans-HC(= S)SH and 2 cc-CH 2 S 2 isomers. The molecular dipole moment is related to charge distribution in molecules, which can influence the intensity of rotational spectrum [11]. Thus, the dipole moments of the five isomers are also collected in Table 1, which have great differences affected by the position of H atoms. The C s point group molecules have a-and b-dipole moment components, and there is a large variation between these two components. In particular, the difference between a and b is about 1.29 D for trans-HC(= S)SH, which is a nearprolate asymmetric rotor with the asymmetry parameter = −0.99(can be seen in Table 2). As expected, the 2ttand 2 cc-CH 2 S 2 isomers have only b components, and the calculated values are 0.2649 D and 2.5602 D, respectively. The dipole moments of cis-HC(= S)SH and 2 cc-CH 2 S 2 are larger than 2.5 D, which lead to more higher intensity of rotational spectrum in the same abundance. Figure 2 gives the intensity of rotational spectrum in the range of 0-3100 GHz at 300 K using the virtual multi-frequency spectrometer [40,41]. And the parameters of the five isomers are taken from B3LYP/Gen = 5 level of theory. It can also be seen in Fig. 2 that the intensity of 2tt-CH 2 S 2 isomer is the weakest, because its dipole moment is the smallest. And it can be predicted that the b-type transitions of trans-HC(= S)SH and 2tt-CH 2 S 2 isomers will be difficult to determine experimentally, which have small b components (0.1623 D and 0.2239 D). Figure 3 gives the relative energies of CH 2 S 2 isomers relative to trans-HC(= S)SH, and they are obtained by total energies (in Table 1) including the harmonic zero-point corrections using CCSD(T)/Gen = Q level of theory. The unit of kilojoule/mole is used for the sake of expressing the relative energy clearly. The order of energy is in agreement with the results from Refs. [13,17,23]: 2tt-CH 2 S 2 > 2tc-CH 2 S 2 > 2 cc-CH 2 S 2 > cis-HC(= S)SH > trans-HC(= S)SH. The most stable isomer is trans-HC(= S)SH as the intramolecular effects of conjugation and hydrogen bonding [14,19]. The second one cis-HC(= S)SH lies energetically about 4 kJ/mol above trans-HC(= S)SH with the CCSD(T)/ Gen = Q level of theory, and the experimental value is 4.2 kJ/mol [23]. The energy difference between DTFA and DTHC is about 140 kJ/mol, while the differences are less than 10 kJ/mol between trans-and cis-HC(= S) SH, or among 2tt-, 2 cc-, and 2tc-CH 2 S 2 . The relatively small differences make them have high sensitivity of temperature [11]. Table 6 Harmonic and fundamental frequencies (in cm − . 1 ) for trans-HC(= S)SH a The results of Ref. [16] were calculated using DZP level of theory and scaled by a factor of 0.90 b The results of Ref. [19] were calculated using SCF-MO/6-31G* level of theory c The values were measured by Ref. [24], while the order was from Ref [19] d The fundamental frequencies were experimental values, and the harmonic data were calculated by SCF/6-31G* with the scaled factor of 0.89 from Ref [18] B3LYP CCSD(T) Pre [16]a Pre [19]b Exp [24]c Ref [18]

Rotational spectra parameters
Several levels of theory are used to calculate the rotational constants for the five isomers which are summarized in Tables 2 and 3, together with the corresponding asymmetry parameters and inertial detects. Rotational constants B and C are predicted well with any level of theory for trans-HC(= S)SH: slightly smaller than experimental values [10]. The opposite is find for the largest A rotational constant; the deviation is about several hundred MHz. That is not surprising, because the experimental uncertainties for rotational constants A are larger than the values for B or C constants by three orders of magnitude. On the other hand, the effect of mixed basis set is significant. In particular, for rotational constants A, the adding of d function on S atoms increases about 171 and 140 MHz using B3LYP and CCSD(T), respectively. The rotational constants of trans-HC(= S)SH well reproduced the most recent experimental values [10] with CCSD(T)/Gen = Q level of theory, which gives the deviation within 0.8%. Thus, the B3LYP/Gen = 5 and CCSD(T)/Gen = Q results of the other four isomers are presented in Table 3. The deviations of the corresponding rotational constants among the five isomers are very tiny, which suggests that the five isomers have similar mass distributions and the position change of H atoms slightly affects the inertia tensor.
Moving to the asymmetry parameter, the five isomers are all near-prolate asymmetric rotors with the K = 2B−A−C A−C = −0.99 , which is very close to the symmetric prolate limit, − 1. As shown in Tables 2 and 3, all the calculated results of DTFA can well reproduce the experimental values [10,22], and the percentage deviations are within 0.08%.
Planar structure of one molecule can be confirmed by the inertial defect, which is defined as Δ = I C − I A + I B [10]. The inertial defect calculated from ground-state rotational constants of the five isomers are given in Tables 2 and 3. The CCSD(T)/Gen = Q level of theory predicts small positive inertial defects, which agree well with planar structures for the five isomers. For the results of B3LYP/Gen = 5, DTFA have planar structure significantly (with the inertial defect values 0.123, 0.186), while the inertial defects of DTHC are slightly larger (0.625, 0.548, 0.576). Table 7 Harmonic and fundamental frequencies (in cm − . 1 ) for CH 2 S 2 isomers. This work, the parameters are determined by B3LYP and CCSD(T) methods employing Gen = 5 and Gen = Q, respectively a The results of Ref. [16] were calculated using DZP level of theory and scaled by a factor of 0.90 b The fundamental frequencies were experimental values, and the harmonic data were calculated by SCF/6-31G* with the factor of 0.89 from Ref [18] c The results of Ref. [19] were calculated using SCF-MO/6-31G* level of theory cis-HC(= S)SH 2tc-CH 2 S 2 2tt-CH 2 S 2 2 cc-CH 2 S 2 B3LYP CCSD(T) Pre [16]a Ref [18]b Pre [19] Quartic and sextic centrifugal distortion constants of a molecule are very important for determining rotational spectra, as molecules at higher rotational energies are not rigid rotors and the vibration-rotation couplings are more significant. The S-reduction is used in this work as the five isomers are all near-prolate asymmetric rotors (see the asymmetry parameters in Tables 2 and 3). The calculated quartic and sextic centrifugal distortion constants (S-reduction) for trans-HC(= S)SH isomer are collected in Table 4 containing experimental data [10]. The effects of basis set extension for trans-HC(= S) SH isomer are clear from cc-pVQZ to cc-pV5Z using B3LYP method: the differences show up in symbol and magnitude. For example, the values of D JK are 247.51 and − 38.77 MHz using B3LYP/cc-pVQZ and B3LYP/ cc-pV5Z, respectively, while the similar results are found between Gen = n and cc-pVnZ (n = Q, 5) basis sets with B3LYP or CCSD(T) method. It suggests that the improvement of basis set can obtain reliable data than adding d function to S atoms for these parameters, which are not the case with rotation constants in Table 2. The differences of quartic centrifugal distortion constants between CCSD(T)/Gen = Q results and experimental values are less than 2.7% except for d 2 , which has a large deviation of 12.5%. This outcome is also considered a good result as the presence of large amplitude motions and the extremely small value of d 2 (about − 3 Hz) [11]. The results of B3LYP/Gen = 5 are similar with the data of CCSD(T)/Gen = Q for the five quartic centrifugal distortion constants. The sextic centrifugal distortion constants of trans-HC(= S)SH isomer are calculated by B3LYP method, and the data of Gen = 5 or cc-pV5Z basis sets are more reliable than the values of n = Q basis sets. Table 5 lists the calculated quartic and sextic centrifugal distortion constants of other isomers determined by B3LYP/ Gen = 5 and CCSD(T)/Gen = Q levels of theory and the experimental data [10] for cis-HC(= S)SH. The quartic centrifugal distortion constants calculated by CCSD(T) is not sensitive to the choice of the structure. A satisfactory agreement for cis-HC(= S)SH isomer can be found in CCSD(T)/ Gen = Q, and the predictions of quartic and sextic centrifugal distortion constants for 2tt-and 2tc-CH 2 S 2 isomers are reliable.  Vibrational spectra parameters Table 6 contains 9 vibrational frequencies of trans-HC(= S) SH isomer, which has 7 of in-molecular-plane A ′ and the remaining 2 of out-of-plane A ε symmetry species. The agreement of fundamental frequencies between B3LYP/ Gen = 5 and experimental ones [18] is excellent with less than 2.2%. The results of B3LYP/Gen = 5 and CCSD(T)/ Gen = Q are consistent within 1.8% for other harmonic frequencies except for 9 The effect of cc-pV(Q + d)Z basis set on S atoms is slightly stronger than basis set improvement from cc-pVQZ to cc-pV5Z. The largest difference between cc-pVQZ and Gen = Q for the 9 harmonic frequencies is about 5.5 cm −1 which appears in the S-H stretching mode 2 . The calculated value 2 of 2655.2 cm −1 in the range of characteristic bond frequencies for S-H stretching is similar to the same vibrations of MgSH [42], AuSH [43], and CuSH [44] molecules, which are 2674.9, 2688, and 2649.9 cm −1 , respectively. Similarly, the characteristic bond frequency 1 of 3085.9 cm −1 for C-H bond stretching is close to the value of 3105.9 cm −1 for trans-HC(= O)OH [45], where it is also adjacent to one single and one double bond. The characteristic bond frequencies 1 and 2 suggest that the two frequencies mainly depend on the binding force of H and other heavy atoms as the small mass of H nucleus. The other thing that we have to notice is that the C = S stretching The intensity of nine frequencies for CH 2 S 2 isomers at the B3LYP/Gen = 5 level of theory mode 4 is the strongest band (see Fig. 4) with the intensity of 112.9 and 135.8 km/mol using CCSD(T)/Gen = Q and B3LYP/Gen = 5, respectively.
The calculated harmonic and fundamental frequencies of cis-HC(= S)SH and DTHC are listed in Table 7 together with previous data for cis-HC(= S)SH isomer [16,18,19]. The harmonic and fundamental frequencies of DTFA are similar to each other, because only the positions of the hydrogens in sulfhydryl are different for these two isomers. The 1 and 2 of two isomers for DTFA are also S-H and C-H stretching characteristic bond frequencies, respectively. The 9 is very sensitive to geometrical isomerism. In the results of CCSD(T)/Gen = Q level of theory, the frequency 9 of trans-HC(= S)SH decreases from 428.7 cm −1 to 360.6 cm −1 of cis-HC(= S)SH. The most intense fundamental frequency v 4 is also sensitive to geometrical isomerism: the B3LYP/Gen = 5 result of v 4 for cis-HC(= S)SH is 23.5 cm −1 higher than that of trans-HC(= S)SH, the corresponding experimental difference is also 23.5 cm −1 [18], which agrees with that the C = S bond length of cis-HC(= S)SH is 0.0013 Å shorter than that of trans-HC(= S)SH. It is similar for v(C = O) stretching frequency with the C = O bond of cis-HC(= O)OH, whose bond length is 0.006 Å shorter than that of trans-HC(= O) OH isomer [45]. The frequencies of v 7 and v 9 of DTFA lie below 475 cm −1 and are not detected so far; thus, the results of this work can be considered valuable prediction.
On the other hand, neither theoretical nor experimental results of harmonic or fundamental frequencies of DTHC exist until now. It can be seen from Table 7 that the B3LYP and CCSD(T) results of harmonic or fundamental frequencies for 2tt-and 2tc-CH 2 S 2 are consistent. Similar to DTFA, 2tc-CH 2 S 2 isomer has seven vibrational modes belonging to A ′ symmetry species and two vibrational modes belonging to A ε species. The most intense frequency for 2tc-CH 2 S 2 (see Fig. 4) is v 2 , which provides a plausible support to detect the spectral feature of 2tc-CH 2 S 2 .
2tt-and 2 cc-CH 2 S 2 isomers belong to the C 2v point group, and their corresponding frequencies lie close to each other. v 1 (totally symmetric specie, A 1 ) and v 7 (anti-symmetric with respect to molecular plane, B 2 ) of 2tt-and 2 cc-CH 2 S 2 are S-H wagging and twisting vibrations, respectively. The fundamental frequency v 7 of 2tt-and 2 cc-CH 2 S 2 is sensitive to geometrical isomerism. The difference of v 7 between 2ttand 2 cc-CH 2 S 2 is about 300 cm −1 using B3LYP/Gen = 5 level of theory, while the differences of v 2 , v 3 , and v 4 between 2tt-and 2 cc-CH 2 S 2 are all less than 10 cm −1 . The values of v 1 − v 7 of 2tt-and 2 cc-CH 2 S 2 isomers are 1.9 cm −1 and 16.7 cm −1 , respectively. The small difference of 1.9 cm −1 suggests that the coupling of two S-H bonds in 2tt-CH 2 S 2 isomer is weaker, which corresponds to longer bond length of S-H (see Fig. 1) [46]. Furthermore, it is worth noting that the calculated IR intensity of 5 is 0, which suggests that the dipole moment in this vibration mode does not change.
The obvious anharmonic effects of DTFA are about 140 and 180 cm −1 , which happened in v 1 (C-H) and v 1 (S-H) (see Tables 6 and 7). All the anharmonic constants of the five isomers are given in Table S2. It can be seen from the B3LYP/Gen = 5 results that the anharmonic effects of DTFA are mainly due to x 11 , x 22 , and x 29 constants, which are all larger than 65 cm −1 . However, the anharmonic effects of v 1 (S-H) and v 7 (S-H) for 2tt-CH 2 S 2 and 2 cc-CH 2 S 2 isomers are about 160 cm −1 and 210 cm −1 , respectively. Similarly, it can be seen from Table S2 that this anharmonic effects of 2tt-CH 2 S 2 and 2 cc-CH 2 S 2 isomers are mainly contributed by x 11 , x 17 , and x 77 constants. The above vibrations all involve the motion of H atoms: C-H or S-H.
The ab initio ro-vibration interaction constants ( a A i , a B i , and a C i , i = 1-9) of the five CH 2 S 2 isomers are given in Table 8. These constants may be perturbed by Coriolis coupling constants; thus, the values of ∑ a B i ∕2 = A e − A 0 are also given in Table 8. Similar expressions also hold for the ∑ a B i and ∑ a C i . As shown in Table 8, the B3LYP and CCSD(T) theory give internal agreements for ∑ a B i and ∑ a C i in DTFA isomers. Thus, the B3LYP/Gen = 5 and CCSD(T)/ Gen = Q data can provide the reliable predictions of spectroscopic constants for CH 2 S 2 isomers. In addition, the cubic and quartic force constants of CH 2 S 2 isomers using B3LYP/Gen = 5 level of theory are shown in Tables S3 and  S4, respectively. These data are calculated in the normal coordinate and can be used for reference.

Conclusion
The equilibrium geometry, dipole moments, total energies, spectroscopic constants, and force fields of five CH 2 S 2 isomers are obtained at B3LYP and CCSD(T) methods with correlation consistent basis sets. Comparing the calculated results with available experimental data, the B3LYP/Gen = 5 and CCSD(T)/ Gen = Q levels of theory can give the reliable prediction. The adding of d function on S atoms has significant effect for A rotational constants (about 150 MHz). The extension from Gen = Q to Gen = 5 with B3LYP method is very essential for the centrifugal distortion constants. The rotational constants of five isomers are similar because the position of H atoms slightly influences moment of inertia, while the position of H significantly affects the bond angles, dipole moments, and molecular point groups of five isomers. Thus, the large variety exists in the intensity of rotational spectra of CH 2 S 2 isomers. The frequencies of CH 2 S 2 isomers have diverse values and symmetry species, which is determined by a molecular point group. The characteristic bond frequencies exist in the C-H and S-H stretching vibrations of DTFA. These new spectroscopic data might be used for the astronomical search in some S-rich prestellar cores and then understand the sulfur chemistry in astrophysical environment.