Spectroscopic constants and anharmonic force fields of F2SO: An ab inito study

Chemical Physics Letters 736 (2019) 136814

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Research paper

Spectroscopic constants and anharmonic force fields of F2SO: An ab inito study

T

Lihan Chi, Meishan Wang , Chuanlu Yang, Xin Li, Xiaoguang Ma ⁎

School of Physics and Optoelectronics Engineering, Ludong University, Yantai 264025, China

HIGHLIGHTS

SO has pyramidal structure and C symmetry. • FSpectroscopic and anharmonic force fields of F SO are predicted. • B3P86/cc-pV5Zconstants and B3P86/MP2 theoretical levels are reasonable to F SO. • 2

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2

2

ARTICLE INFO

ABSTRACT

Keywords: Anharmonic force field Spectroscopic constant DFT MP2 Thionyl fluoride

The equilibrium geometry, spectroscopic constants and anharmonic force field of F2SO ( X A ) have been reported employing B3LYP, B3P86, B3PW91 and MP2 methods combining with Dunning’s correlation-consistent cc-pVQZ, cc-pV5Z, aug-cc-pVQZ and aug-cc-pV5Z basis sets. The calculated equilibrium geometry, equilibrium rotational constants and fundamental vibrational frequencies of F2SO well reproduce the previous theoretical or experimental values. The calculated equilibrium geometry verifies that F2SO has pyramidal structure and Cs symmetry instead of C1 symmetry. The ground-state rotational constants, harmonic vibrational frequencies, anharmonic constants, quartic and sextic centrifugal distortion constants, cubic and quartic force constants, vibration-rotation interaction constants, and Coriolis coupling constants of F2SO are also predicted. The calculated results show that B3P86 and MP2 methods are superior to B3LYP and B3PW91 methods for the spectroscopic constants of F2SO, we hope that our predictions can provide the useful data for the experiment study of the corresponding spectroscopic constants of F2SO.

1. Introduction Thionyl fluoride (F2SO) is an important degradation product of Sulfur hexafluoride (SF6), which can be produced from the break-up of SF6 near high-voltage power lines or lightning in the thunderstorms in atmosphere [1–4]. SF6 has been widely used as an insulator in switchgear and circuit breakers, because of its arc-quenching properties [5]. However, the release of SF6 into the atmosphere is considered to contribute to global warming [6,7]. When partial discharge (PD), a spark, an arc, or overheating happens in SF6 gas, SF6 will decompose and produce SF4, SF3, SF2, S2F10, or some other low-fluoride sulfides, which have very active chemical properties and react with impurities H2O or O2. The decomposition products of SF6 conclude SOF2, SOF4, SO2F2, SO2, HF, and H2S [8–11]. The study of the decomposition products of SF6 is closely relevant to the electronics industry and in the reduction of greenhouse gases [6,7]. Therefore, the gas-phase

⁎

1

spectroscopic characterization of F2SO and other related molecules has aroused much concern in recent years. The rotational spectroscopy of F2SO can reveal its internal structure and physicochemical properties. In 1953, Ferguson measured the microwave spectrum of F2SO in the 16–36 GHz range and concluded that F2SO had Cs symmetry and pyramidal structure for the first time [12]. In the subsequent years, the numerous K-doubling transitions and the extended measurements of F2SO into the 40–70 GHz region was recorded as well as the centrifugal distortion parameters were determined by Yamaguchi et al. [13] and Dubrelle & Destombes [14]. Rimmer et al. measured rotational spectra of F2SO in the ν4 = 1 and ν6 = 1 vibrationally-excited states [15]. Lucas & Smith reported the rotational spectroscopy of F2SO in the rage of 8.6–77.1 GHz with R-branch lines up to J = 5 and K-doubling transitions up to J = 40 with estimated uncertainties of 50 kHz, meanwhile, they obtained the transition frequency and accurate quartic distortion constants of F2SO [16]. Stone et al. determined g factors, magnetic susceptibilities, and other

Corresponding author. E-mail address: [email protected] (M. Wang).

https://doi.org/10.1016/j.cplett.2019.136814 Received 3 September 2019; Received in revised form 30 September 2019; Accepted 30 September 2019 Available online 01 October 2019 0009-2614/ © 2019 Elsevier B.V. All rights reserved.

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Table 1 The contraction of basis sets of F, S, O for F2SO. Atoms O F S Atoms O F S

Table 2 The equilibrium geometry of F2SO.

cc-pVQZ Before 12s6p3d2f1g 12s6p3d2f1g 16s11p3d2f1g

After 5s4p3d2f1g 5s4p3d2f1g 6s5p3d2f1g

cc-pV5Z Before 14s8p4d3f2g1h 14s8p4d3f2g1h 20s12p4d3f2g1h

After 6s5p4d3f2g1h 6s5p4d3f2g1h 7s6p4d3f2g1h

aug-cc-pVQZ Before 13s7p4d3f2g 13s7p4d3f2g 17s12p4d3f2g

After 6s5p4d3f2g 6s5p4d3f2g 7s6p4d3f2g

aug-cc-pV5Z Before 15s9p5d4f3g2h 15s9p5d4f3g2h 21s13p5d4f3g2h

After 7s6p5d4f3g2h 7s6p5d4f3g2h 8s7p5d4f3g2h

Method/basis sets

r(SeO)Å r(SeF)Å α(FeSeO)° α(FeSeF)° τ(F.S.F.O)°

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ MP2/aug-cc-pV5Z Theo. Ref. [16] rs Theo. Ref. [34] rm(1) Expt. Ref. [3] r0 Expt. Ref. [33] Theo. Refs. [3,33] re

1.4262 1.4211 1.4262 1.4211 1.4225 1.4176 1.4224 1.4177 1.4241 1.4189 1.4240 1.4188 1.4256 1.4206 1.4265 1.4210 1.4159 1.4143 1.4389 1.420 1.426

1.6114 1.6047 1.6121 1.6046 1.5985 1.5921 1.5989 1.5919 1.6018 1.5952 1.6022 1.5950 1.5945 1.5876 1.5958 1.5879 1.5868 1.58357 1.5715 1.583 1.613

106.5421 106.6356 106.5435 106.639 106.478 106.6118 106.4751 106.6226 106.5251 106.6322 106.5232 106.6373 106.6611 106.7396 106.6119 106.7125 106.66 106.635 106.12 106.2 106.52

93.0377 93.0561 93.0339 93.0661 93.0031 93.0388 92.9876 93.0565 93.0840 93.1010 93.0728 93.1180 92.7587 92.7757 92.6791 92.7545 92.79 92.965 94.044 92.2 93.03

108.2517 108.3709 108.2521 108.3783 108.163 108.3363 108.1544 108.3553 108.2465 108.3818 108.2405 108.3936 108.303 108.4033 108.2176 108.3636 108.312 108.513 109.11 107.572 108.223

comprehensive database for this relevant degradation product and can promote air pollution prevention. The paper is organized as follows: in Section 2, the calculation methods are briefly reviewed. In Section 3, results and discussion are presented. Finally, Section 4 closes with the conclusions.

Fig. 1. The geometry of F2SO in C1 symmetry.

2. Calculation methods The anharmonic force field of a molecule can be calculated by analyzing the second derivative of energy with respect to cartesian coordinates and transforming it to mass-weighted coordinates according to the Gaussian 09 W suit of programs [19]. Since this transformation can only be used for the calculation of stable points, the geometry of F2SO molecule must be optimized first. Based on the equilibrium geometry of F2SO, the anharmonic field and other spectral constants of F2SO are calculated at the same level of theory. B3LYP, B3P86, B3PW91, and MP2 methods are used in this paper [20–24]. DFT method is developed on the basis of Hohenberg-Kohn theory, which uses the electron density function to describe and determine the properties of the system without resorting to the wave function of the system. MP2 method is a high accuracy quantum chemical calculation method, in which electron correlation is considered using RayleighSchrodinger’s perturbation theory. We use cc-pVnZ and aug-cc-pVnZ basis sets (n = Q and 5) to calculate the equilibrium geometry of F2SO [25,26]. The polarization functions are concluded in these basis sets, the contractions of basis sets for F, S, O [27] are shown in Table 1. In a subsequent step, the harmonic vibrational frequencies are then derived via analytical second derivative methods. We evaluated the cubic force constants by carrying out the numerical differentiation of the analytical hessians computed at points displaced along the normal coordinates, following the procedure described by Schneider and Thiel [28]. By using these cubic force constants in normal coordinates, in conjunction with the equilibrium rotational constants and Coriolis coupling constants, one can obtain the sextic centrifugal distortion constants employing the expressions available in the literature [29–31].

Fig. 2. The geometry of F2SO in Cs symmetry.

magnetic properties of F2SO by a Zeeman study [17]. In 2002, Tucceri et al. obtained the optimized geometry, harmonic vibrational frequencies, and enthalpies of formation of F2SO using ab initio molecular orbital theory and density functional theory (DFT) calculation [18]. In 2018, Keogh et al. measured the sub-millimeter/THz spectrum of F2SO in the frequency range 286–508 GHz, which involved the two-hundred fifty transitions frequencies in the range J = 16 to J = 30, with Kc = 0–28, each with a c-type pattern. They determined the optimized geometry, dipole moments, rotational constants, quartic and sextic centrifugal distortion constants [3]. However, so far the harmonic vibrational frequencies, ground-state rotational constants, vibration-rotation interaction constants, anharmonic constants, force constants, Coriolis coupling constants of F2SO have not been reported theoretically or experimentally. In this paper, density functional theory (DFT: B3LYP, B3P86, B3PW91) and second-order Moller-Plesset perturbation (MP2) methods are employed to investigate the anharmonic force fields and spectroscopic constants of F2SO. The equilibrium geometries, rotational constants, fundamental vibrational frequencies and centrifugal distortion constants of F2SO are obtained and compared with the previous experimental or theoretical values. On this basis, the vibrationrotation interaction constants, anharmonic constants, force constants, Coriolis coupling constants of F2SO are reasonably predicted, which will provide the reference data for the theoretical or experimental research of F2SO in the future. This work is a good contribution to building a

3. Results and discussion 3.1. Equilibrium geometry According to the experimental data of geometry of F2SO given by Keogh et al. [3]: bond length r(1Se3O) = 1.438(9) Å and r (1Se2F) = 1.5715 Å, bond angle α(2Fe1Se3O) = 106.12° and 2

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Table 3 Equilibrium and ground-state rotational constants of F2SO (MHz). Method/basis sets

Ae

Be

Ce

A0

B0

C0

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ Expt. Ref. [3] Theo. Ref. [16]

8313.260 8384.638 8306.795 8385.384 8441.048 8512.469 8439.291 8512.301 8402.511 8475.075 8399.168 8475.680 8512.304 8587.384 8504.218 8614.810 8614.73

8177.606 8235.401 8173.674 8236.548 8266.92 8321.868 8264.897 8322.266 8242.951 8300.391 8240.296 8302.359 8240.464 8300.439 8227.514 8356.942 8356.99

4817.861 4852.174 4814.691 4852.501 4885.681 4917.427 4884.818 4916.896 4864.334 4898.339 4862.712 4898.704 4896.681 4933.226 4893.33 4952.946 4952.88

8258.431 8328.874 8251.562 8329.552 8387.137 8457.476 8385.138 8457.255 8348.273 8419.987 8344.653 8420.543 8455.776 8530.336 8447.649

8137.246 8194.428 8132.965 8195.511 8227.581 8281.845 8225.369 8282.199 8203.218 8260.138 8200.338 8262.054 8199.825 8259.086 8186.311

4794.946 4828.917 4791.540 4829.198 4863.233 4894.661 4862.232 4894.105 4841.767 4875.526 4839.987 4875.858 4873.264 4909.332 4869.383

Table 4 Fundamental vibrational frequencies of F2SO (cm−1). Method/basis set

ν1

ν2

ν3

ν4

ν5

ν6

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ Expt. Ref. [16] Expt. Ref. [35]

1315.013 1322.803 1311.796 1321.886 1336.648 1344.020 1334.537 1343.030 1331.382 1339.840 1328.652 1339.262 1351.716 1356.93 1345.180 1335.0 1333

775.090 774.171 767.025 772.779 795.601 797.930 791.580 797.464 789.862 792.621 785.609 791.597 796.529 799.577 789.067 808.2 808

708.519 704.437 701.715 703.489 730.732 731.199 726.502 731.289 725.117 725.785 720.458 725.183 732.795 733.446 724.448 747.0 747

504.441 509.398 503.073 509.184 514.901 520.021 514.360 520.564 513.081 518.308 512.247 518.329 519.513 525.064 516.696 530.4 530

377.865 381.508 377.104 381.094 384.070 387.546 383.870 388.325 383.410 386.820 382.788 386.620 386.846 390.304 384.408 392.5 393

349.068 353.605 347.866 353.338 357.271 362.028 356.956 362.553 355.629 360.473 354.949 360.301 367.588 372.127 365.638 377.8 378

Table 5 Harmonic vibrational frequencies of F2SO (cm−1). Method/basis set

ω1

ω2

ω3

ω4

ω5

ω6

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ

1330.792 1338.453 1327.811 1337.713 1350.629 1357.938 1348.376 1356.977 1345.221 1353.766 1342.683 1353.284 1363.064 1368.361 1356.624

783.356 784.968 777.866 783.743 806.156 808.676 801.996 807.607 801.482 803.994 796.948 802.971 808.006 810.909 800.72

716.754 715.884 710.502 715.072 742.239 742.577 737.611 741.969 737.463 737.749 732.440 737.212 744.864 745.758 736.782

510.331 515.706 509.254 515.575 520.61 525.938 520.012 525.779 518.855 524.290 518.119 524.159 525.427 530.594 522.577

381.010 384.336 380.383 384.254 386.957 390.137 386.640 390.086 386.255 389.477 385.860 389.399 390.048 393.534 387.737

354.103 359.025 353.138 359.037 362.361 367.292 361.953 367.254 360.662 365.683 360.069 365.668 372.881 376.870 370.954

α(2Fe1Se4F) = 94.044°, dihedral angle τ(2Fe1Se4Fe3O) = 109.11° (as shown in Fig. 1), we can easily figure out that bond angle α(4Fe1Se3O) is equal to 107.10641°, which shows that F2SO satisfies the C1 symmetry. However, the previous literatures [12–16] make clear that F2SO has Cs symmetry. In other words, when the bond angle α(4Fe1Se3O) = α(2F-1S-3O) = 106.12° and α(2Fe1Se4F) = 94.044°, the dihedral angle τ(2Fe1Se4Fe3O) of F2SO should be equal to 108.06946° instead of 109.11° measured by Keogh et al. (as shown in

Fig. 2). Therefore, in the following calculation, we used the geometry data of F2SO molecule corresponding to Fig. 2 within the constraint of Cs point group symmetry. The analytical gradient method [19,32] is used in optimizing the equilibrium geometry of F2SO to find the lowest energy structure near the starting structure by solving the first derivative, and obtaining the harmonic vibrational frequency by analyzing the second derivative. The calculated equilibrium geometry of F2SO using different theoretical 3

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Table 6 Quartic centrifugal distortion constants of F2SO (KHz).

Table 8 Anharmonic constants of F2SO (cm−1).

Method/basis set

ΔJ

ΔJK

ΔK

δJ

δK

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ Expt. Ref. [3] Expt. Ref. [16]

6.203 6.257 6.232 6.262 6.093 6.143 6.111 6.147 6.095 6.142 6.116 6.146 6.082 6.136 6.126 4.543 4.586

2.659 2.470 2.633 2.463 2.806 2.621 2.782 2.613 2.759 2.587 2.737 2.586 2.197 2.092 2.210 −2.433 −2.475

−7.845 −7.710 −7.843 −7.707 −7.907 −7.774 −7.898 7.770 −7.862 −7.737 7.856 −7.740 −7.309 −7.254 −7.353 9.978 10.005

0.142 0.151 0.140 0.151 0.164 0.175 0.165 0.175 0.155 0.165 0.154 0.163 0.238 0.223 0.214 1.540 1.550

174.286 157.268 177.675 157.662 136.974 124.309 136.628 124.678 148.780 135.041 149.364 136.177 81.196 77.657 80.435 2.246 2.225

xij

x11 x12 x13 x14 x15 x16 x22 x23 x24 x25 x26 x33 x34 x35 x36 x 44 x 45 x 46 x55 x56 x 66

levels are given in Table 2. According to the existed data of the rs [16], re [3,32], and experimental structure [33] of F2SO shown in Table 2, the dihedral angle τ(F.S.F.O) of F2SO can be determined 108.312°, 108.223° and 107.572°, respectively. It can be found that the calculated r(SeO) and r(SeF) diminish with the basis sets increasing, however, a(FeSeO), a(F-S-F), and τ(F.S.F.O) increase. The addition of a diffuse functions (aug-) to the cc-pVnZ basis set has little effect on the final result of F2SO. The deviation of the calculated r(SeO) and r(SeF) at B3LYP/cc-pV5Z theoretical level with the experimental values [3] are −0.0178 Å and 0.0332 Å, respectively. The r(SeO) and r(S-F) deviation for B3P86/ B3PW91/MP2 methods with cc-pV5Z basis set is within −0.0213 Å/ −0.02 Å/−0.0183 Å and 0.0206 Å/0.0237 Å/0.0161 Å respectively. The calculated a(FeSeO), a(FeSeF), and τ(F.S.F.O) by B3LYP/cc-pV5Z theoretical level reproduces the experimental value within 0.5156°, −0.9879° and 0.3014°, respectively. The deviation of the calculated a (FeSeO), a(FeSeF), and τ(F.S.F.O) at B3P86/cc-pV5Z theoretical level with the experimental values [3] are 0.4918°, −1.0052° and 0.26684°, respectively. The result for B3PW91 and MP2 methods with cc-pV5z basis set is within 0.5122°, −0.943°, 0.1234° and 0.6196°, −1.2683°, 0.3338° respectively. The discrepancy between the MP2/cc-pV5Z result and the structure parameters [33] is only 0.0006 Å, 0.0046 Å, 0.5396°, 0.5715° and 0.8313° for r(SeO), r(SeF), a(FeSeO), a(FeSeF), and τ(F.S.F.O) respectively. Compared with the experimental results [3,33], the theoretical values obtained by this work are more consistent with

B3P86

B3PW91

MP2

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

−6.593 1.322 0.367 −1.315 −1.771 −0.192 −1.870 −7.251 −3.494 −0.986 −3.222 −1.561 −3.150 −3.122 −3.616 −0.458 −0.620 −1.008 0.204 1.541 −0.921

−6.645 1.381 0.329 −1.203 −1.507 −0.255 −1.879 −7.379 −3.684 −1.015 −3.278 −1.558 −3.348 −2.865 −3.261 −4.830 −0.582 −1.084 −0.081 1.113 −0.940

−6.469 1.195 0.288 −1.354 −1.772 −0.160 −2.089 −8.116 −3.545 −1.075 −3.342 −1.749 −3.314 −3.038 −3.522 −0.467 −0.541 −0.931 −0.160 1.377 −0.872

−6.558 1.251 0.273 −1.322 −1.560 −0.225 −2.040 −8.320 −3.564 −0.977 −3.261 −1.700 −3.400 −2.778 −3.192 −0.496 −0.608 −1.085 −0.107 1.075 −0.933

−5.088 0.900 −0.087 −1.120 −1.940 −0.020 −2.047 −7.878 −3.499 −0.987 −3.304 −1.622 −3.440 −2.833 −3.412 −0.483 −0.644 −1.117 −0.252 1.006 −0.935

−5.259 1.184 −0.056 −1.024 −1.960 −0.024 −2.049 −8.154 −3.491 −0.862 −3.144 −1.670 −3.486 −2.833 −3.314 −0.427 −0.513 −0.838 −0.301 0.906 −0.780

the experimental results [33]. The calculated two bond lengths of F2SO are of high accuracy, and the calculated bond angle and dihedral angle errors are also within a reasonable range. 3.2. Rotational constants The equilibrium rotational constants(Ae, Be, Ce)and ground-state rotational constants(A0, B0, C0)of F2SO are calculated employing DFT and MP2 methods combined with Dunning correlation-consistent basis sets, which are shown in Table 3. The experimental results recorded by Keogh et al. [3] and the theoretical results obtained by Lucas and Smith [16] are also listed in Table 3. In the resent work, the MP2/aug-cc-pV5Z result is not considered because this theoretical level is beyond our computational ability. By taking into account the effects of vibrationrotation coupling via perturbation theory [23,24], the ground-state rotational constants can be obtained by the corresponding equilibrium rotational constants, the specific expression is as follows. B i

B0 = Be i

vi +

1 +… 2

(1)

Table 7 Sextic centrifugal distortion constants of F2SO (Hz). Method/basis set

ΦJ

ΦJK

ΦKJ

ΦK

φJ

φJK

φK

B3LYP/cc-pVQZ B3LYP/cc-pV5Z B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pV5Z B3P86/cc-pVQZ B3P86/cc-pV5Z B3P86/aug-cc-pVQZ B3P86/aug-cc-pV5Z B3PW91/cc-pVQZ B3PW91/cc-pV5Z B3PW91/aug-cc-pVQZ B3PW91/aug-cc-pV5Z MP2/cc-pVQZ MP2/cc-pV5Z MP2/aug-cc-pVQZ Expt. Ref. [3] Expt. Ref. [3] Expt. Ref. [16]

0.0041 0.0013 0.0041 0.0012 −0.0002 −0.0022 −0.0005 −0.0022 −0.0011 −0.0010 −0.0009 −0.0010 −0.0062 −0.0089 −0.0082 0.00201

−2.2238 −1.8240 −2.2892 −1.8872 −1.8532 −1.5751 −1.8425 −1.5816 −2.1206 −1.7995 −2.1297 −1.8218 −0.8582 −0.6385 −0.6693 −0.11111 0.01595 0.15450

7.2292 6.1303 7.4438 6.1490 6.0565 5.1619 6.0242 5.1839 6.9254 5.8904 6.9581 5.9642 2.8354 2.1346 2.2304 0.17718

−5.0115 −4.2518 −5.1608 −4.2651 −4.2050 −3.5865 −4.1831 −3.6019 −4.8080 −4.0919 −4.8313 −4.1433 −1.9729 −1.4898 −1.5548 0.07610

0.0119 0.0104 0.0122 0.0104 0.0085 0.0075 0.0085 0.0075 0.0094 0.0083 0.0095 0.0083 0.0048 0.0041 0.0043

−2.2609 −1.8240 −2.3357 −1.8223 −1.6361 −1.3212 −1.6184 −1.3232 −1.9262 −1.5552 −1.9290 −1.5713 −0.6653 −0.4924 −0.5446

−0.17600

0.07240

0.03030

−0.11180

−375.1228 −274.8168 −398.0294 −277.0368 −167.1271 −124.0602 −166.0502 −125.2263 −217.3527 −161.5878 −220.2240 −165.8087 −33.7766 −31.3432 −35.3219 0.06683 0.06763 0.14150

0

4

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theoretical level and the experimental data [3] are 2.67%, 1.45%, and 2.03%, respectively. The corresponding deviations at B3PW91/cc-pV5Z theoretical level are 1.62%, 0.68%, and 1.10%, respectively. The calculated values using B3P86 and MP2 methods are more accurate than the B3LYP and B3PW91 methods. At the theoretical level of B3P86/ccpV5Z, the difference of the equilibrium rotational constants of F2SO between the calculated and the experimental value are 1.19%, 0.42%, 0.72%, respectively. When using aug-cc-pV5Z basis set, the difference are 1.19%, 0.41%, 0.73%, respectively. One can clearly see that we obtain almost the same results employing cc-pV5Z and aug-cc-pV5Z basis sets. At the theoretical level of MP2/cc-pV5Z, the difference of the equilibrium rotational constants of F2SO between the calculated and the experimental value are 0.32%, 0.68%, 0.40%, respectively. Although we don’t give the MP2/aug-cc-pV5Z results due to the limitation of calculation conditions, we believe that the calculated MP2/cc-pV5Z results are basically consistent with MP2/aug-cc-pV5Z results. Through the above analysis, one can draw the conclusion that the calculated MP2/cc-pV5Z results are reliable. Therefore, the values of ground-state rotational constants of F2SO at this theoretical level can be used as a prediction.

Table 9 Vibration-rotation interaction constants of F2SO (MHz). Constant

A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 C 1 C 2 C 3 C 4 C 5 C 6

B3P86

B3PW91

MP2

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

23.600

23.808

23.702

23.849

21.525

21.949

24.592

25.044

24.683

24.967

27.335

27.591

27.676

28.326

27.861

28.315

30.519

31.076

10.870

11.221

11.176

11.487

11.293

11.412

−64.675

−70.792

−61.393

−67.140

−99.339

−101.423

85.759

92.381

82.449

88.699

121.721

123.490

19.913

20.017

20.038

20.110

17.752

17.775

18.718

18.987

18.814

18.965

20.361

20.771

22.182

22.569

22.392

22.625

24.189

24.460

13.660

14.001

13.957

14.275

13.806

14.056

−27.293

−29.487

−26.008

−28.084

−40.249

−41.552

31.499

33.961

30.274

32.617

45.420

47.198

1.338

1.365

1.328

1.337

1.053

1.167

0.872

1.351

0.863

1.310

1.642

2.279

24.239

24.066

24.410

24.138

25.237

25.088

2.680

3.021

2.733

3.056

3.280

3.472

−13.722

−15.204

−13.022

−14.411

−20.947

−22.539

29.488

30.934

28.821

30.195

36.569

38.320

3.3. Fundamental and harmonic vibrational frequencies F2SO molecule has six vibrational modes, the corresponding fundamental vibrational frequencies are SO stretching mode ν1, SF2 symmetrical stretching mode ν2, SF2 asymmetrical stretching mode ν3, SF2 waging mode ν4, FSO bending mode ν5 and SF2 bending mode ν6. Table 4 lists the calculated fundamental vibrational frequencies of F2SO, along with the available experimental data [16,35]. The deviations between the experimental values [16] and the calculated values at B3LYP/cc-pV5Z theoretical level for fundamental vibrational frequencies of F2SO are 0.91%, 4.21%, 5.70%, 3.60%, 2.80%, and 6.40%, respectively. At the same time, the corresponding deviations are 0.68%,

where vi is quantum number of vibrational mode i and is the vibration-rotation interaction constant which is given in Table 9. The similar expression can be used for A and C, where the summation is over the all vibrational modes of F2SO. It can be seen from Table 3 that the values of both the equilibrium rotational constant and the groundstate rotational constant increase with the increase of the basis set and are closer to the experimental values. The deviations between the equilibrium rotational constants of the F2SO at the B3LYP/cc-pV5Z B i

Table 10 Cubic force constants of F2SO (cm−1). Constant

f111 f221 f222 f331 f332 f411 f421 f422 f433 f441 f442 f444 f531 f532 f543 f551 f552 f554 f611 f621 f622 f633 f641 f642 f644 f653 f655 f661 f664 f666

B3P86

B3PW91

MP2

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

−428.630 22.336 −151.390 33.163 −179.495 16.572 16.141 35.578 41.788 21.210 −21.633 43.832 32.367 22.534 −17.949 60.002 −6.833 40.841 −6.302 −8.553 20.055 −1.524 1.307 −7.567 5.041 −6.260 −30.775 −2.303 30.587 76.818

−430.852 23.184 −152.778 34.582 −182.402 16.531 16.268 36.721 43.364 21.336 −22.419 44.884 32.780 22.532 −18.203 60.452 −6.826 41.053 −6.176 −8.863 19.922 −2.221 1.107 −7.737 5.139 −5.665 −31.717 −2.259 31.640 76.874

−429.385 22.202 −150.890 33.087 −179.115 16.948 15.868 35.779 42.543 21.000 −21.742 44.249 31.993 22.876 −18.131 59.216 −6.936 40.936 −6.465 −8.375 20.093 −0.935 1.406 −7.523 5.045 −6.313 −30.544 −2.323 30.186 76.590

−432.002 22.979 −151.782 34.410 −181.466 16.702 16.102 36.663 43.805 21.160 −22.464 45.185 32.537 22.658 −18.319 59.831 −6.936 41.168 −6.287 −8.710 19.988 −1.638 1.178 −7.710 5.132 −5.691 −31.506 −2.261 31.267 76.736

−399.017 22.709 −160.712 33.778 −188.729 16.048 15.946 38.867 45.329 21.454 −24.746 45.046 32.502 22.693 −18.776 59.791 −7.490 39.893 −10.589 −9.617 20.486 −2.243 0.813 −8.041 5.319 −5.695 −31.551 −3.200 32.579 75.430

−403.375 23.549 −162.802 35.075 −192.378 16.036 16.017 39.899 46.358 21.382 −25.431 45.076 32.665 22.775 −19.050 59.852 −7.714 39.586 −10.312 −9.773 20.291 −3.186 0.675 −8.176 5.462 −5.119 −31.975 −3.024 33.347 75.500

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where i is the ith harmonic vibrational frequency and x ij is an anharmonic constant. According to Eq. (2), fundamental vibrational frequencies (Table 4), and anharmonic constants (Table 7), we can deduce harmonic vibrational frequencies of F2SO. So far, there are no experimental and theoretical reports on harmonic frequencies of F2SO. The calculated of harmonic vibrational frequencies of F2SO are shown in Table 5. According to the above analysis, the harmonic vibrational frequencies of F2SO calculated at B3P86/cc-pV5Z and MP2/cc-pV5Z theoretical levels are reliable.

Table 11 Quartic force constants of F2SO (cm−1). Constant

f1111 f 2111 f2211 f2221 f2222 f3311 f3321 f 3322 f3333 f4111 f4211 f4221 f4222 f4331 f4332 f4411 f4421 f4422 f4433 f4441 f4442 f4444 f5311 f5322 f5333 f5511 f5521 f5522 f5533 f5541 f5544 f5553 f5555 f6111 f6211 f6221 f6222 f6331 f6411 f6422 f6433 f6442 f6551 f6552 f6554 f6611 f6622 f6633 f6642 f6644 f6662 f6664 f6666

B3P86

B3PW91

MP2

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

122.472 −1.494 −6.121 −3.695 24.072 −5.882 −6.235 39.691 42.206 −6.815 −10.324 1.482 −12.893 0.520 −13.057 −18.008 2.769 1.349 −2.059 −3.326 −0.814 0.733 −13.390 −6.343 −10.189 −31.216 1.485 −0.989 1.029 −3.706 4.882 2.304 15.139 2.841 1.020 2.102 −8.762 1.431 2.552 2.866 −1.446 −1.314 2.104 1.763 −2.830 −0.615 −5.862 −10.832 −2.828 1.821 −8.637 8.087 14.096

122.739 −1.521 −6.260 −3.848 24.814 −5.845 −6.382 41.159 43.952 −6.734 −10.352 1.486 −13.217 0.564 −13.335 −17.814 2.849 1.215 −2.328 −3.569 −0.939 0.643 −13.227 −6.213 −10.102 −30.387 1.874 −1.097 0.635 −3.986 5.148 3.789 17.495 2.830 1.066 2.168 −8.828 1.440 2.768 2.847 −1.630 −1.557 2.525 1.275 −2.599 −0.832 −6.058 −11.098 −3.085 1.797 −8.497 8.064 13.597

126.240 −1.536 −6.277 −3.524 20.615 −6.121 −6.115 36.473 39.396 −6.996 −10.194 1.210 −12.270 0.209 −12.611 −17.676 2.597 0.835 −2.475 −3.716 −0.683 0.730 −13.283 −5.735 −9.576 −30.918 1.213 −1.391 0.683 −4.114 5.266 2.653 15.770 2.998 0.842 2.266 −9.137 1.611 2.583 3.190 −1.010 −1.698 2.362 1.241 −2.342 −0.488 −6.353 −10.960 −2.822 2.092 −9.160 8.597 14.799

126.091 −1.502 −6.252 −3.490 21.889 −5.958 −6.030 38.318 41.321 −6.733 −10.270 1.419 −12.481 0.470 −12.712 −17.651 2.831 1.126 −2.394 −3.5992 −0.721 0.535 −13.328 −5.721 −9.560 −30.528 1.958 −1.057 0.781 −3.897 5.114 3.801 17.073 2.865 1.042 2.174 −8.975 1.459 2.688 2.939 −1.439 −1.547 2.437 1.270 −2.597 −0.721 −6.001 −10.836 −2.980 1.754 −8.544 8.072 13.677

114.863 2.330 −5.554 −5.063 27.951 −5.569 −7.959 44.585 48.498 −6.941 −9.636 1.650 −14.220 1.145 −14.602 −15.911 2.349 1.851 −1.573 −3.119 −1.094 0.782 −12.540 −6.748 −11.494 −29.962 1.387 −0.923 0.790 −3.443 4.572 2.510 14.363 3.525 0.750 2.256 −8.735 1.358 2.555 2.850 −1.911 −1.381 1.776 1.654 −2.747 −0.526 −5.494 −11.114 −2.563 1.860 −8.106 7.701 12.611

115.586 1.972 −5.173 −5.103 29.262 −5.463 −8.069 46.038 49.577 −6.595 −9.579 1.990 −14.399 1.467 −14.584 −15.755 2.449 2.703 −1.389 −2.571 −0.722 1.675 −12.816 −7.008 −11.620 −30.265 1.439 −0.531 0.644 −2.698 4.975 1.919 13.818 3.223 1.026 2.018 −8.471 0.999 2.694 2.871 −2.025 −1.027 1.157 2.347 −2.733 −0.322 −4.733 −10.995 −2.485 3.170 −7.588 8.212 14.995

3.4. Quartic and sextic centrifugal distortion constants Considering the fact that the chemical bond of a real molecule is not actually rigid, the nuclear separation will be subjected to the centrifugal force when the molecule vibrates, which will lead to the increase of bond length, thus resulting in centrifugal distortion effect. The molecular centrifugal distortion effect can be described by the centrifugal distortion constants, which include the quartic, sextic, and other higher order centrifugal distortion constants. The calculated quartic centrifugal distortion constants of the equilibrium state and the experimental data [3,16] of ground-state of F2SO are given in Table 6. The quartic centrifugal distortion constants of F2SO measured in KHz are much smaller than the rotational constants (103 MHz), so they affects the molecular rotation much weaker than the rotational constants. It can be seen from our data that the calculated results at different theoretical levels are very close, but there are two points worth noticing. The first is that ΔJK is negative in the experiment, while the calculated value at all theoretical levels is positive, while the situation is the opposite for ΔK. The second is that the error between the theoretical values and experimental values of ΔJ, ΔJK, ΔK and δJ are within the allowable range, but it is surprised that the calculated value of δK is more than 35–60 times as high as the experimental value. Therefore, in order to solve this problem, the further precise measurements of quartic centrifugal distortion constant of F2SO from the related experiment are necessary. The sextic centrifugal distortion constants of F2SO are much smaller than quartic centrifugal distortion constants, which are given in Table 7 along with the relevant experimental data [3,16]. It can be seen that the sign and size of the sextic centrifugal distortion constants of F2SO at different methods are not exactly same. The calculated value of ΦJ using the B3LYP method is positive, however, it is negative when B3P86, B3PW91, and MP2 methods are used. The calculated value of φK using B3LYP/cc-pV5Z theoretical level is −274.817 Hz, nevertheless, it is only −31.343 Hz when MP2/ cc-pV5Z theoretical level is considered. The prominent differences are also existed between the calculated value and the experimental value for the sextic centrifugal distortion constants φK of F2SO. The calculated value of φK at MP2/ccpV5Z theoretical level is −31.34316 Hz, however, the experimental value [3] is only 0.06683 Hz. Such errors are intolerable at all times. We look forward to the experimental and theoretical efforts to solve it in the future.

1.27%, 2.12%, 1.20%, 1.26%, and 4.17% for B3P86/cc-pV5Z theoretical level; 0.36%, 1.92%, 2.84%, 2.28%, 1.45%, and 4.59% for B3PW91/ cc-pV5Z theoretical level; 1.64%, 1.07%, 1.81%, 1.01%, 0.56%, and 1.50% for MP2/cc-pV5Z level. Comparing with the results of cc-pVnZ (n = Q, 5), the use of diffuse functions (aug-) has little influence on values of fundamental vibrational frequencies of F2SO. Therefore, B3P86/cc-pV5Z and MP2/cc-pV5Z theoretical levels can provide the better fundamental vibrational frequencies for F2SO. The vibration of a real molecule is anharmonic, therefore, it is necessary to take the anharmonic correction into account. The fundamental vibrational frequency of F2SO can obtained by anharmonic constant and corresponding harmonic vibrational frequency:

vi =

i

+ 2x ii +

1 2

x ij j i

3.5. Anharmonic constants Table 8 shows the calculated anharmonic constants of F2SO using the B3P86, B3PW91 and MP2 methods combined with the cc-pVQZ and cc-pV5Z basis sets. Since the calculated accuracy of B3LYP method is obviously lower than that of B3P86, B3PW91 and MP2 methods as well as aug-cc-pVnZ basis sets have little influence on the final results, so B3LYP method and aug-cc-pVnZ basis sets are not considered in the discussion of the other spectral constants subsequently. It is worth noting that the calculated values of x13 using B3P86 and B3PW91 methods are positive, while the calculated values using the MP2 method are negative and obviously different from the values given by B3P86 and B3PW91 methods. The other anharmonic constants of F2SO at different theoretical levels are relatively close and the symbols are

(2) 6

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Table 12 Coriolis coupling constants of F2SO. Constant

( ) 21 ( ) 31 ( ) 32 ( ) 41 ( ) 42 ( ) 43 ( ) 51 ( ) 52 ( ) 53 ( ) 54 ( ) 61 ( ) 62 ( ) 63 ( ) 64 ( ) 65 ( ) 21 ( ) 31 ( ) 32 ( ) 41 ( ) 42 ( ) 43 ( ) 51 ( ) 52 ( ) 53 ( ) 54 ( ) 61 ( ) 62 ( ) 63 ( ) 64 ( ) 65 ( ) 21 ( ) 31 ( ) 32 ( ) 41 ( ) 42 ( ) 43 ( ) 51 ( ) 52 ( ) 53 ( ) 54 ( ) 61 ( ) 62 ( ) 63 ( ) 64 ( ) 65

B3P86

B3PW91

MP2

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

cc-pVQZ

cc-pV5Z

−0.10466

−0.10468

−0.10521

−0.10512

−0.10539

−0.10824

0.07149

0.07183

0.07201

0.07214

0.07866

0.08283

−0.51463

−0.51356

−0.51798

−0.51637

−0.51367

−0.50971

−0.13578

−0.13563

−0.13546

−0.13537

−0.13648

−0.14047

0.09431

0.09387

0.09449

0.09406

0.09203

0.09453

0.41897

0.42297

0.41573

0.41996

0.42667

0.42526

0.11476

0.11673

0.11483

0.11663

0.11740

0.12658

−0.29683

−0.29960

−0.29682

−0.29952

−0.29966

−0.30347

0.05761

0.05749

0.05745

0.05736

0.05793

0.05961

−0.37447

−0.37264

−0.37369

−0.37224

−0.36988

−0.36617

0.01383

0.01476

0.01343

0.01441

0.01641

0.01753

−0.05400

−0.05475

−0.05386

−0.05461

−0.05490

−0.05715

0.70845

0.70635

0.70803

0.70622

0.70311

0.70507

−0.05759

−0.05770

−0.05728

−0.05743

−0.05678

−0.05885

−0.35234

−0.35236

−0.35330

−0.35303

−0.35386

−0.35872

−0.32078

−0.32085

−0.32248

−0.32221

−0.32302

−0.32533

−0.34910

−0.34712

−0.35084

−0.34844

−0.34846

−0.34390

0.05186

0.05068

0.05166

0.05060

0.04697

0.04531

−0.41618

−0.41571

−0.41520

−0.41491

−0.41832

−0.42218

0.28907

0.28770

0.28960

0.28829

0.28206

0.28411

0.29274

0.29020

0.29427

0.29160

0.28628

0.28278

−0.56243

−0.56387

−0.56170

−0.56330

−0.56415

−0.56113

0.19103

0.19091

0.18994

0.19004

0.18431

0.18413

0.17658

0.17621

0.17608

0.17582

0.17755

0.17915

−0.08166

−0.08720

−0.07998

−0.08559

−0.09100

−0.09541

0.04239

0.04525

0.04118

0.04416

0.05028

0.05270

−0.16550

−0.16782

−0.16508

−0.16738

−0.16828

−0.17178

−0.35554

−0.36017

−0.35374

−0.35853

−0.36133

−0.36154

−0.17653

−0.17687

−0.17557

−0.17602

−0.17403

−0.17689

0.44674

0.44431

0.44793

0.44533

0.44654

0.44037

−0.36212

−0.36220

−0.36404

−0.36373

−0.36465

−0.35989

0.28858

0.28673

0.28997

0.28781

0.28595

0.28596

0.10280

0.10353

0.10394

0.10441

0.10685

0.11235

−0.46982

−0.46928

−0.46871

−0.46838

−0.47224

−0.46703

0.32633

0.32478

0.32693

0.32544

0.31841

0.31429

−0.38041

−0.37932

−0.38083

−0.37968

−0.37691

−0.38353

0.46506

0.46576

0.46439

0.46529

0.46581

0.46918

−0.08343

−0.08253

−0.08247

−0.08178

−0.07666

−0.07518

0.19933

0.19891

0.19877

0.19848

0.20043

0.19818

0.18056

0.18494

0.17885

0.18340

0.18751

0.19638

0.04785

0.05108

0.04648

0.04986

0.05676

0.05830

−0.18683

−0.18945

−0.18636

−0.18895

−0.18997

−0.19002

0.11020

0.11491

0.10873

0.11349

0.11687

0.11476

−0.19928

−0.19966

−0.19819

−0.19870

−0.19645

−0.19568

−0.29391

−0.29175

−0.29468

−0.29246

−0.29329

−0.29019

consistent. The calculated results indicate that anharmonic constants have obvious influence on the interactions between different vibrational modes of F2SO.

B3P86, B3PW91 and MP2 methods with the cc-pVQZ and cc-pV5Z basis sets are given in Table 9. It can be seen that the calculated values at the above different theoretical levels are relatively close. Since there are no experimental or theoretical data on the vibration-rotation interaction constants of F2SO up to now, we expect that our calculated results can provide theoretical references.

3.6. Vibration-rotation interaction constants The vibration-rotation interaction constants of F2SO employing 7

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L. Chi, et al.

3.7. Cubic and quartic force constants [2]

The quadratic force constant corresponds to the harmonic vibrational frequency of F2SO, which has been given in Table 5. A series of spectral constants can be obtained by the cubic force field of F2SO, including vibration-rotation interaction constants, Coriolis coupling constants and centrifugal distortion constants. The most of calculated cubic force constants of the F2SO in normal coordinates are listed in Table 10, which are very close at different theoretical levels. The part of calculated values of the quartic force constant are listed in Table 11. It's amazing that the calculated values of f 2111 using B3P86 and B3PW91 methods are negative, while the calculated values using the MP2 method are positive. The values of quartic force constants from the DFT methods are relatively close to each other. To our knowledge, the cubic and quartic force constants are not reported theoretically or experimentally to date, so we expect that the calculated cubic and quartic force constants of F2SO can be used for reference.

[3] [4] [5] [6] [7] [8] [9] [10]

3.8. Coriolis coupling constants

[11]

Table 12 lists the Coriolis coupling constants of F2SO. The calculated results are fairly similar at different theoretical levels at present work. Thus, the current results for Coriolis coupling constants can be responsibly regarded as the prediction values for F2SO.

[12] [13] [14]

4. Conclusion

[15]

The equilibrium geometry, spectroscopic constants, and anharmonic force field of F2SO have been studied with the B3LYP, B3P86, B3PW91, and MP2 methods employing cc-pVnZ, aug-cc-pVnZ (n = Q, 5) basis sets. For F2SO, the calculated accuracy of B3LYP method is obviously lower than that of B3P86, B3PW91 and MP2 methods, meanwhile, augcc-pVnZ basis sets have little influence on the final results of F2SO. The calculated equilibrium geometry, equilibrium rotational constants, and fundamental vibrational frequencies of F2SO are in good agreement with the experimental values, which guarantee that the calculated harmonic vibrational frequencies and the ground-state rotational constants of F2SO are reliable. In addition, the quartic and sextic centrifugal distortion constants, anharmonic constants, vibration-rotation interaction constants, Coriolis coupling constants, and force constants of F2SO with B3P86, B3PW91 and MP2 methods are calculated and discussed, which can be regarded as the prediction values for F2SO.

[16]

Declaration of Competing Interest

[26]

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[27]

[17]

[18] [19] [20] [21] [22] [23] [24] [25]

[28] [29]

Acknowledgements

[30] [31] [32]

This work was supported by the National Natural Science Foundation of China (Grant No. 11474142) as well as the Taishan Scholars project of Shandong Province (ts2015 11055). All calculations were carried out at the Langchao Super Computer Center (LCSCC) of Ludong University.

[33] [34] [35]

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