Multichannel RRKM-TST and Direct-Dynamics CVT Study of the

Article pubs.acs.org/JPCA

Multichannel RRKM-TST and Direct-Dynamics CVT Study of the Reaction of Hydrogen Sulfide with Ozone S. Hosein Mousavipour,* Maryam Mortazavi, and Omid Hematti Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran S Supporting Information *

ABSTRACT: The kinetics of the reaction of ozone with hydrogen sulfide was studied theoretically. High-level ab initio calculations were carried out to build the potential energy surface. The mechanism of the title reaction was found to be much more complicated than what is reported in the literature to date. According to our results, six different chemically activated intermediates are involved along the proposed mechanism on its lowest singlet potential energy surface that play an important role in the kinetics of this system. Multichannel RRKM-TST and CVT calculations have been carried out to compute the temperature dependence of the individual rate constants for different channels and also the overall rate constant for the consumption of the reactants. The major products are sulfur dioxide and water at lower temperatures, in good agreement with experimental reports, while at higher temperatures, formation of the other products like O2, H2SO, and radicals like cis/trans-HOSO, SH, HO3, and OH also become important.

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and Toby6,7 studied the kinetics of the title reaction over the temperature range of 298−343 K and the pressure range of 0.005−0.1 Torr in O3 and 0.2−5 Torr in H2S. They found that O2 is the most abundant product of the reaction and that the ratio of (O2 formed)/(O3 used) approaches a value of 1.5. They stated that SO2, H2O, and H2S (produced from the SH + SH reaction) are the products of the reaction. Under a high concentration of H2S, the rate of the reaction is reported to be of (1.75 ± 0.25) order in O3 concentration. They suggested the following rate constant expression for the disappearance of the reactants: k = 1.6 × 109 exp(−21.7 kJ mol−1/RT) L mol−1 s−1. Despite the important role of the H2S + O3 reaction on the chemistry of the atmosphere, no accurate data on the kinetics and mechanism of the title reaction are reported in the literature to date. This is the purpose of the present study to investigate the detailed reaction mechanism for the title reaction and calculate the rate constants for the individual steps and also the overall rate constant for the disappearance of the reactants by means of the multichannel RRKM-TST method12,13 and canonical variational transition-state theory (CVT).

INTRODUCTION The effect of different chemicals present in the stratosphere on the depletion of the ozone layer has been the subject of numerous investigations. One of the most abundant sulfurbased species in the atmosphere is the hydrogen sulfide. The atmospheric residence time of hydrogen sulfide is less than one day.1 Hydrogen sulfide is mainly produced by volcanoes, mineral water springs and organic decomposition in swamps, the seaside, and oceans. Other sources of hydrogen sulfide are paper industries, oil and gas refineries, and chemical fertilizer factories. These different sources of hydrogen sulfide make it one of the potential pollutants in the atmosphere.1,2 The effect of hydrogen sulfide on the depletion of stratospheric ozone is of major interest. SO2 and water have been suggested to be the major products of oxidation of H2S by ozone.3−7 SO2 itself is one of the major pollutants in the atmosphere. Presumably, a small fraction of SO2 dissolved in water to form H2SO38−11 produces acid rain. Gregor and Martin3 suggested that sulfur dioxide and water are formed in equal amounts when hydrogen sulfide reacts with ozone. Cadle and Ledford4 studied the reaction of H2S + O3 in a flow system and reported that the rate of O3 disappearance is of 3/2 order in O3 concentration and zero order in H2S, under high H2S concentration conditions. They declared that the reaction was in part heterogeneous. A study by Hales et al.5 on the kinetics of H2S + O3 in a flow system led to the rate expression for the disappearance of the reactants of r = 3.8 × 105[O3]1.5[H2S]0.5 mol L−1 s−1. They disputed the heterogeneity of this system that was reported by Cadle and Ledford4 but accepted the simple stoichiometry of the reaction. Glavas © XXXX American Chemical Society

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COMPUTATIONAL METHOD Gaussian03 program14 was used to optimize the geometries and calculate the energies of the stationary points. The geometries of the stationary points were optimized at the MP215/Aug-ccReceived: May 14, 2013 Revised: July 8, 2013

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dx.doi.org/10.1021/jp404738d | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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pVTZ16,17 level. Single-point CCSD(T)18−20/Aug-cc-pVTZ// mp2/Aug-cc-pVTZ calculations were carried out to obtain more accurate energies of the stationary points along the PES. Harmonic vibrational frequencies were obtained at the vibration energies (zero-point energy, ZPE). The intrinsic reaction coordinate (IRC) was calculations21 at the MP2/631+G(d,p) level in order to characterize the stationary points as local minima or first-order saddle points and obtain the zeropoint have been utilized to validate the connection of each transition state to the corresponding minima along the reaction paths. The calculated relative energies at different levels of theory and ZPEs are listed in Table 1. Harmonic vibrational

Geometries and Potential Energy Surfaces. On the basis of the results from the previous studies that have been reported in the literature and our theoretical calculations, the suggested mechanism for the title reaction is shown in Scheme 1. Figure 1 shows the optimized structures of the stationary points listed in Scheme 1. The geometric parameters in Figure 1 are compared with corresponding values in the literature, where available.24−30 The geometrical structures of all of the stationary points in Z-matrix format are provided as Supporting Information. A schematic of the potential energy surface (PES) for the reaction of H2S with O3 at the CCSD(T)/Aug-cc-pVTZ level is shown in Figure 2. In Scheme 1, the energized species are marked with a character “*”. The relative energies quoted in Figure 2 are corrected for the ZPEs. As shown in Scheme 1, the suggested mechanism of the title reaction is much more complicated than what was reported in the literature.3−7 Sulfur dioxide, water, and molecular oxygen have been suggested as the major products for this system3−7 (the products of reactions R5, R7, R9, and R11−R14). The first step in our suggested mechanism is the association process to produce a van der Waals like structure vdw1 (Figures 1 and 2) that converts to vibrationally excited intermediate Int1 by passing over the transition state TS1 with a 54.6 kJ mol−1 barrier height (reaction R1) or converts to HO3 and SH through transition state TS10 with a 100.8 kJ mol−1 barrier height (reaction R10). Chemically activated intermediate Int1 is 117.8 kJ mol−1 more stable than the reactants at the CCSD(T)/Aug-cc-pVTZ level. The energized intermediate Int1 undergoes a stabilization process via collisions or converts to a more stable energized intermediate Int2 (158.2 kJ mol−1 more stable than Int1) by passing over the transition state TS2 with a 31.6 kJ mol−1 barrier height. The energized intermediate Int2 experiences the stabilization process or undergoes a rearrangement process to form another energized intermediate Int3 (sulfonic acid like structure). Intermediate Int3 undergoes a rearrangement process to form Int4 via transition state TS4, which ends at the energized trans- or cis-HOSO* plus OH radicals, or a dissociation process to form SO2 and H2O (reaction R5), or a rearrangement process to form vibrationally energized sulfurous acid, H2SO3* (reaction R6). According to our calculations, H2SO3 is 63.2 kJ mol−1 more stable than sulfonic acid (Int3) and 600.5 kJ mol−1 more stable than the reactants. The highly vibrationally energized H2SO3 dissociates to SO2 and H2O via transition state TS7 (reaction R7) or to cisor trans-HOSO plus OH radical via reaction R8. Voegele et al.31 suggested that sulfurous acid, H2SO3, the chemically activated product of reaction R6, is thermodynamically unstable, which under standard conditions is expected to easily dissociate into the SO2 and H2O (reaction R7). They reported that the half-life of sulfurous acid is only 24 h at room temperature and suggested a value of −33.8 kJ mol−1 at the G2(MP2) level of theory for the ΔE0 of reaction R7. Li and McKee32 reported values of −24.2 and 125.4 kJ mol−1 for the standard reaction energy (ΔE0) and activation energy of dissociation of H2SO3 into the SO2 and H2O at the G2 level of theory, respectively. Our results at the CCSD(T)/Aug-ccpVTZ level indicate that SO2 + H2O is 11.9 kJ mol−1 more stable than H2SO3. Our results indicate that the potential energy barrier for dissociation of sulfurous acid into the sulfur dioxide and water is 104.4 kJ mol−1 at the CCSD(T)/Aug-ccpVTZ level, about 21 kJ mol−1 less than the value reported by Li and McKee.

Table 1. Relative Energies of Various Species at Different Levels of Theory in kJ mol−1 on the Singlet Surface species

MP2a

B3LYPb

CCSD(T)a

ZPEc

reactants TS1 TS2 TS3 TS4 TS5 TS6 TS7 TS9 TS10 TS11 TS12 + 1O2 TS13 + 1O2 TS15 + OH Int1 Int2 Int3 Int4 Vdw1 cis-HO3 + SH trans-HO3 + SH H2SO3 cis-HOSO + OH trans- HOSO + OH HSO2 + OH cis-HSOH + 1O 2 trans-HSOH + 1O2 H2OS + O2 H2SO + 1O2 Vdw2 1 O2 + SH + OH SO2 + H2O 1 SO + 2OH 3 SO + 2OH

0.0 75.9 −96.9 −31.2 −86.8 −382.8 −373.8 −523.4 −116.1 58.3 80.2 162.6 114.5 3.6 −89.9 −268.3 −579.0 −317.9 −7.3 123.7 126.2 −628.7 −283.6 −272.5 −173.6 −68.1 −94.0 60.0 −8.7 −18.7 228.0 −645.4 108.9 15.3

0.0 40.5 −95.8 −24.5 −72.1 −294.8 −272.3 −453.6 −138.9 43.6 100.9 163.8 90.3 −61.4 −135.3 −250.7 −470.1 −331.2 −9.7 −7.6 −11.2 −558.6 −310.3 −298.6 −172.3 −81.3 −108.5 25.0 1.8 −9.4 154.7 −578.3 44.8 −73.8

0.0 52.0 −101.9 −56.2 −105.9 −355.5 −352.0 −505.9 −136.2 98.7 98.8 141.0 85.4 −66.9 −134.4 −295.7 −564.2 −346.1 −6.65 38.2 43.7 −624.5 −318.0 −308.3 −198.3 −97.5 −123.0 22.3 −30.2 −38.2 169.2 −626.9 47.6 −48.0

56.1 58.7 71.8 69.5 66.7 69.9 70.1 65.9 71.7 58.2 65.7 55.7 49.9 52.5 72.7 75.7 83.0 79.2 60.2 58.2 58.8 80.0 63.7 61.6 63.4 62.5 65.4 72.8 60.6 63.3 45.9 70.6 49.0 49.0

a

Along with the Aug-cc-pVTZ basis set. bAlong with the 6-311+G (d, p) basis set. cCalculated at the B3LYP/6-311+G(d,p) level and scaled by a factor of 0.96.

term values are scaled by a factor of 0.9622 and, along with moments of inertia for all of the stationary points, are listed in Table 2. The CCSD(T)/Aug-cc-pVTZ//MP2/Aug-cc-pVTZ T1 diagnostic23 is carried out for all of the saddle point geometries. The T1 diagnostic is a measure of the importance of the multireference effect. The value of T1 diagnosis for all of the saddle point structures was found to be equal to or less than 0.03, indicating that multireference effects are minor. B



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Table 2. Harmonic Vibrational Wave Numbers (cm−1) and Moments of Inertia (amu Å2) for Various Species at the B3LYP/6311+G(d,p) Level species H2S O3 vdw1 vdw2 TS1 TS2 TS3 TS4 TS5 TS6 TS7 TS9 TS10 TS11 TS12 TS13 TS15 Int1 Int2 Int3 Int4 (HO)2SO cis-HO3 cis-HOSO trans-HSOH H2SO H2OS HSO2

I1, I2, I3

wave numbers 2699.3, 2169.5, 2614.4, 2285.0, 2803.3, 3896.5, 3947.4, 3627.0, 3898.6, 3925.0, 3950.3, 3799.2, 2633.0, 2552.7, 3651.5, 3648.1, 1936.1, 3567.6, 3640.9, 3642.7, 3640.7, 3594.4, 3567.8, 3532.0, 3676.8, 2393.4, 3731.2, 2227.9,

2679.9, 1119.6, 2599.2, 2280.5, 1755.0, 2654.6, 2592.0, 1986.2, 2138.1, 2376.6, 2055.6, 1702.0, 2235.8, 2418.1, 1988.1, 1986.2, 1140.8, 2514.9, 2283.2, 2454.6, 3637.5, 3591.6, 1346.6, 1300.4, 2622.0, 2392.8, 3638.5, 1507.0,

1171.2, 716.9, 1168.9, 1165.7, 1130.9, 721.9, 237.4, 190.6, 136.8, 79.6, 59.6, 43.2 1517.1, 1181.5, 1011.2, 976.3, 934.7, 205.7, 111.2, 72.3, 52.6, 33.9 1308.6, 1277.0, 1186.2, 858.1, 676.5, 580.4, 414.5, 334.1, 241.8, 840.5i 1472.9, 1169.0, 1128.8, 1067.5, 597.1, 555.6, 504.2, 336.7, 182.0, 262.6i 1275.9, 1141.8, 1068.9, 1013.1, 845.8, 450.4, 376.5, 362.9, 257.9, 1006.0i 1100.8, 1029.8, 828.9, 728.7, 680.0, 430.8, 366.7, 266.0, 174.7, 1349.9i 1465.0, 1243.3, 1185.0, 856.8, 723.4, 561.0, 493.7, 361.1, 294.4, 1890.2i 1370.1, 1169.6, 1136.4, 891.2, 648.1, 566.5, 434.8, 412.5, 196.4, 1921.5i 1366.7, 1329.8, 1094.4, 895.1, 765.8, 557.2, 521.2, 493.3, 258.7, 1758.7i 1426.4, 1193.6, 1085.3, 728.5, 627.1, 603.3, 380.2, 260.6, 185.0, 2037.7i 1211.2, 1121.0, 993.1, 637.9, 490.3, 364.3, 234.7, 167.0, 90.9, 1010.3i 1179.1, 979.6, 916.8, 742.1, 703.8, 622.9,395.1, 337.7, 238.6, 347.5i 1003.4, 877.1, 509.9, 1530.1i 1002.4, 876.3, 509.4, 1516.1i 795.0, 618.5, 390.5, 1554.2i 1362.2, 964.6, 895.5, 697.6, 572.3, 479.1, 425.6, 349.0, 299.5, 108.7 1310.0, 1118.1, 1070.2, 935.2, 818.0, 551.1, 405.2, 292.3, 181.3, 142.8 1334.5, 1116.2, 1090.3, 1041.9, 988.5, 746.3, 537.4, 429.6, 383.8, 207.3 1244.5, 1111.0, 733.3, 719.5, 550.9, 489.1, 365.0, 333.3, 170.3, 129.4 1162.9, 1042.6, 1012.8, 688.1, 676.9, 447.3, 407.9, 391.8, 303.9, 153.2 1189.1, 665.9, 418.3, 172.4 1002.3, 726.5, 338.5, 185.5 1157.8, 995.9, 754.1, 457.1 1211.7, 1074.7, 1023.4, 962.9 1549.7, 697.8, 646.5, 432.3 1095.0, 1084.8, 816.7, 422.7

3.5, 1.9, 1.6 43.0, 38.2, 4.8 240.3, 206.0, 43.3 256.5, 240.7, 19.6 151.7, 124.6, 37.3 188.9, 170.7, 23.0 132.4, 110.7, 25.2 138.1, 124.2, 23.3 110.9, 67.0, 55.3 102.5, 59.9, 54.4 104.1, 74.4, 49.9 128.5, 117.5, 32.8 213.1, 180.7, 42.4 125.2, 92.3, 40.4 43.9, 43.5, 1.8 43.9, 43.5, 1.8 66.6, 57.0, 12.2 158.9, 142.0, 30.6 140.8, 122.7, 32.2 105.1, 59.4, 57.2 146.1, 128.0, 35.5 107.5, 67.6, 61.0 51.3, 44.0, 7.3 69.6, 54.8, 14.8 35.3, 34.4, 2.5 27.6, 27.2, 3.2 47.2, 46.4, 1.7 62.6, 54.5, 10.1

thioperoxide, which is formed via transition state TS12, undergoes a rearrangement process to form less stable H2OS (thiooxonium) (reaction R13) or dissociates to SH + OH (reaction R14). Our results indicate that H2OS is 64.7 kJ mol−1 less stable than H2SO at the CCSD(T) level. The structure of hydrogen thioperoxide has been studied theoretically and experimentally.26,37 In this system, the contribution of reaction R12 (formation of HSOH from H2SO) is negligible because of the high activation energy of TS11. The energy changes during the interaction of H2S with ozone in the lowest triplet state were also studied. Figure 3 shows the lowest triplet PES for the title reaction, and Table 3 lists the relative energies and ZPEs. On the triplet surface, the association reaction of H2S + 3O3 produces energized intermediate Int5 that is 299.8 kJ mol−1 more stable than the reactants (its optimized structure at the MP2 level is shown in Figure 1). Highly energized intermediate Int5 dissociates into H2SO plus triplet molecular oxygen, which is 3.4 kJ mol−1 less stable than Int5. Rate Constants Calculations. As shown in Figure 2, the singlet PES consists of four energized intermediates Int1−Int4 plus energized sulfurous acid and hydrogen thioperoxide. For those channels that proceed via the formation of chemically activated species (reactions Rw1, Rw2, Rw3, Rw4, R4′, R5, Rw6, R7, R8, and R9 and the overall rate constant for disappearance of the reactants), a method based on RRKM calculations with steady-state assumption (RRKM-TST) for the activated species (according to a method suggested by Dean38) was used to estimate the corresponding rate constants. To locate the position of the bottlenecks for those channels with

Furthermore, the produced HOSO* in reaction R8 rearranges to HSO2 (reaction R15) or dissociates to 1SO + OH (reaction R16) or to 3SO + OH (reaction R16′). The total energy of 1SO + 2OH radicals is 641.0 kJ mol−1 above the energy of sulfurous acid and 40.5 kJ mol−1 less stable than the reactants, while the total energy of 3SO + 2OH radicals is 545.4 kJ mol−1 above the energy of sulfurous acid and 55.1 kJ mol−1 more stable than the reactants; therefore, the contribution of these channels and also stabilization process Rw8 (the stabilization process of HOSO*) are negligible in this system and in our formulation of the RRKM-TST method to calculate the rate constants for channels R1−R9 are neglected. Hydrogen trioxide (HO3), the product of reaction R10, has long been postulated as a key intermediate in important atmospheric processes but has proved difficult to detect.33 Despite some experimental and theoretical reports, there is still significant uncertainty regarding the bond dissociation energy of HO3 → O2 + OH and as to which conformer of HO3 (cis or trans) is most stable. Varandas has studied the structural properties of HO3 and its dissociation energy in detail.34−36 Our results indicate that the cis conformer of HO3 is 4.5 kJ mol−1 more stable than its trans conformer at the CCSD(T)/ Aug-cc-pVTZ//CISD/Aug-cc-pVDZ level. Another path for the title reaction is the formation of van der Waals vdw2 that is 31.0 kJ mol−1 more stable than the reactants with a 108.4 kJ mol−1 activation energy. vdw2 converts to H2SO + O2 (reaction R11), which is 25.7 kJ mol−1 more stable than the reactants. H2SO undergoes a rearrangement process to produce chemically activated trans- or cis-HSOH (reaction R12). HSOH, either known as oxadisulfane or hydrogen C



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Multichannel RRKM-TST method was used to calculate the individual rate constants for those channels whose chemically activated intermediates are involved (channels Rw1, Rw2, Rw3, Rw4, R4′, R5, Rw6, R7, R8, and R9). The formulation for this method is based on the idea suggested by Dean38 to assume steady-state approximation for the formation and consumption of the energized intermediates in a complex reaction. Assuming steady-state approximation for the energized intermediates in the title reaction leads to the expressions for the second-order rate constants of different channels. The lifetime of each energized intermediate was assumed to be long enough to make the energy redistribution statistical. In Scheme 2, k(Rwx)’s are the stabilization rate constants for the stabilization of the corresponding energized intermediates as shown in Scheme 1, k(Rx)’s are the rate constants for the corresponding channels, and k(loss) is the rate constant for consumption of the reactants. Γ is the tunneling factor, Be is the ratio of the electronic partition functions, h is the Planck constant, Q#a is the product of translational and rotational partition functions for TS1, QH2S and QO3 are the products of the reactant partition functions, G(E+) is the sum of vibrational states of TS1 at the internal energy E+, kx(E)’s are the microcanonical rate coefficient for the corresponding step in the energy range of E+ to E+ + dE+, which is calculated from the quotient of the sum of states to the density of states of the corresponding step, and ω(=Zβc[A]) is the collisional stabilization rate constant for the energized intermediates, where βc is the collision efficiency. D−D6 are defined as

Scheme 1

D = w + k −1(E) + k 2(E) D1 = w + k −2(E) + k 3(E) + k 9(E) D2 = w + k −3(E) + k4(E) + k5(E) + k6(E) D3 = w + k −4(E) + k4′(E) D 4 = w + k −6(E) + k 7(E) + k 8(E)

no saddle point, the microcanonical variational RRKM39 calculations have been carried out (for reactions R4′, R8, R14, R16, and R16′). For the channels in reactions R10, R11, R12, R13, R14, R15, R16, and R16′, CVT was used to predict the rate expressions. For CVT calculations, the GAUSSRATE9.140 program, which is an interface between POLYRATE9.3.141 and Gaussian03, was used. The results from the CCSD(T)/Aug-cc-pVTZ level calculations were used to calculate the rate constants. A standard RRKM program by Zhu and Hase42 was employed to calculate the sum and density of states of the intermediates and transition states. The necessary data for RRKM calculations are summarized in Tables 1 and 2. In RRKM calculations, a step size of ΔE+ = 4.2 kJ mol−1 was used to span the available energy for the activated complex, and the external rotations were treated as being adiabatic. The ratio of the electronic partition functions was assumed to be equal to 1. N2 was chosen as the bath gas using an expression of 400 cm−1 (T/298)0.8 for ⟨ΔEdown⟩.43,44 The sum of states was calculated according to the Tardy et al.45 method, in which only a fraction of the ZPE (aEz) was included in the classical energy at each point along the reaction coordinate. For those channels with no saddle point (reactions R4′, R8, R14, and R16), microcanonical variational RRKM calculation was carried out to locate the position of the bottlenecks. Tables 4−7 show the results of the microcanonical variational RRKM calculations for channels R4′, R8, R14, and R16, respectively. Typically, a tight bottleneck was found for these reactions.

D5 = D2 × D3 × D 4 − k −4(E) × k4(E) × D 4 − k −6(E) × k6(E) × D3 D6 = D1 × D5 − k −3(E) × k 3(E) × D3 × D 4

In calculating the rate constants introduced in Scheme 2, the tunneling process is just important for reaction R1 and its reverse reaction. For channels R2, R4, R6, and R7, the available energy of the corresponding starting species is high enough to cause these species to surmount the corresponding barrier heights, and the tunneling process should not be important. In order to calculate the tunneling factor, it is assumed that a particle with an energy of E approaches an unsymmetrical Eckart barrier. In this method, the tunneling factor can be calculated as46 ⎛ V ⎞ Γ = exp⎜ ⎟ ⎝ kBT ⎠

∫0

∞

⎛ E ⎞ dE κ(E) exp⎜ − ⎟ ⎝ kBT ⎠ kBT

(1)

where V is the effective barrier height corrected for the ZPE and κ(E) is the transmission probability for a particle with the energy E approaching an Eckart barrier. To calculate the tunneling factor, a numerical integration program from Brown47 was used. Γ in eq (1) is related to the barrier heights for the forward and reverse reaction and frequency of an imaginary vibration, ν*, in a well created by inverting the barrier. To calculate the value of Γ, αi (=2πVi/hν*, where i is 1 D



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Figure 1. continued

E



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Figure 1. Optimized geometries of the stationary points at the MP2/Aug-cc-pVTZ level. The values in parentheses are from (a) ref 24, (b) ref 25, (c) refs26 and 27, (d) ref 28, (e) ref 29, and (f) ref 30.

or 2 for forward or reverse reaction, respectively) and u* (=hν*/kBT) were needed as input. The multichannel RRKM rate constants for different channels were calculated according to the equations in Scheme 2, and the results as Arrhenius plots for reactions Rw1, Rw2, Rw3, Rw4, R4′, R5, Rw6, R7, R8, R9, and R(loss) at 760 Torr of pressure are shown in Figure 4. As shown in Figure 4, almost the same values were found for the rate constants k(R9) and k(loss). In Figure 4, our calculated k(loss) is compared with values reported by Glavas and Toby6,7 and Becker et al.48 and DeMore et al.49 at 300 K. Becker et al. and DeMore et al. in two different review studies suggested that the overall rate constant for consumption of the reactants should be less than 12.0 L mol−1 s−1 at 300 K, in good agreement with our k(loss). To the best of our knowledge, no available rate constant data for the other channels quoted in Figure 4 is reported in the literature. Nonlinear least-squares fitting to the predicted rate constants at

Figure 2. Schematic of the PES of the reaction of H2S + 1O3 at the CCSD(T)/Aug-cc-pVTZ level. The energies are corrected for ZPEs.

F



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k w6 = 5.3 × 10−2 × T 0.99 exp( −49.1 kJ mol−1/RT)

k 7 = 12.8 × T 2.19 exp( −48.6 kJ mol−1/RT) k 8 = 1.1 × T 2.77 exp( −47.6 kJ mol−1/RT) k 9 = 5.3 × 105 × T1.64 exp( −48.9 kJ mol−1/RT)

kloss = 4.6 × 105 × T1.66 exp( −48.8 kJ mol−1/RT)

As shown in Figure 4, reaction R9 (formation of SO2 + H2O) is the major channel on the singlet surface at lower temperatures. Reactions R5 and R7 (formation of SO2 + H2O) besides R8 are the next important channels in this systems, indicating that the major products in our calculations are the same as what is reported in the previous studies except for the formation of HOSO, the product of channel R8. Stabilization rate constants are not important in this system, which means experimental detection of sulfonic acid and sulfurous acid in this system should be difficult.

Figure 3. Schematic of the PES of the reaction of H2S + 3O3 at the CCSD(T)/Aug-cc-pVTZ level. The energies are corrected for ZPEs.

Table 3. Relative Energies of Various Species at Different Levels of Theory in kJ mol−1 on the Triplet Surface species 3

H2S + O3 Int5 H2SO + 3O2

MP2a

B3LYPb

CCSD(T)a

ZPEc

0.0 −339.9 −335.2

0.0 −251.7 −250.0

0.0 −310.6 −306.2

50.9 61. 7 60.6

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VARIATIONAL TRANSITION-STATE THEORY CALCULATIONS Computational Details. As shown in Figure 2, because of higher activation energies, contribution of channels R10, R11, R12, R13, R14, R15, R16, and R16′ in this system are negligible; therefore, we did not include their rate constants in our RRKM-TST calculations. Channels R10, R11, R12, R13, and R15 proceed by passing over the corresponding saddle points, TS10, TS11, TS12, TS13, and TS15, respectively. No saddle point was found for reactions R14 and R16, where the microcanonical RRKM method was used to locate the position of the bottlenecks. Their rate constants were calculated using the POLYRATE9.3.1 program. Dual-level CVT was used to calculate the rate constants for these channels. The CVT rate constant, kCVT, might be calculated at temperature T by minimizing the generalized TST rate constant, kGT(T,s), as a function of s

a

Calculated at the Aug-cc-pVTZ basis set. bCalculated at the 6-311+G (d, p) basis set. cCalculated at the B3LYP/6-311+G(d,p) level and scaled by a factor of 0.96.

Table 4. Microcanonical Variational RRKM Results for Unimolecular Dissociation Reaction R4′ E(ν,j)a 24.7 29.4 33.1 49.8 58.2 66.5 74.9 91.6 100 108

E#(ν,j)b R#(Ȧ )c 2.8 5.7 11.4 28.2 36.7 45.1 53.6 70.4 78.9 87.0

3.11 3.04 2.89 2.88 2.87 2.86 2.85 2.84 2.83 2.82

E0d

N(ν,j)e × 104

G(E#)f × 104

k(E) × 1011(1/s)

21.9 21.8 21.7 21.6 21.5 21.4 21.3 21.2 21.1 21.0

0.00162 0.00342 0.00623 0.05342 0.13120 0.29670 0.62700 2.41000 4.41500 7.80500

0.00183 0.00831 0.05547 2.54600 9.81100 31.0600 85.4700 480.900 1025.00 2062.00

0.3379 1.0231 2.6711 14.290 22.411 31.390 40.870 59.830 69.580 79.220

kCVT(T ) = min kGT(TS) s

=σ

The total energy available to the system in kJ mol−1. bTransition-state energy in kJ mol−1. cPosition of the bottleneck in Å. dClassical energy difference between the reactant and the transition state. eDensity of states in cm−1. fSum of states.

⎡ −V (sCVT) ⎤ KBT QGT(T , sCVT) ⎥ exp⎢ MEP R h KBT Q (T ) ⎣ ⎦

a

where s is the arc length along the minimum-energy path (MEP) measured from the saddle point, sCVT is the value of s at which kGT(T,s) has a minimum, σ is the reaction path degeneracy, KB and h are the Boltzmann and Planck constants, respectively, VMEP(sCVT) is the classical MEP potential at s = sCVT, and QGT(T,sCVT) and QR(T) are the internal (rotational, vibrational, and electronic) partition functions of the generalized transition state at s = sCVT and the reactants, respectively. Normally, a generalized normal-mode analysis projecting out frequencies at each point along the path is performed50 to calculate both the vibrational partition function along the MEP and the ground-state vibrationally adiabatic potential curve

the CCSD(T) level in Figure 4 gave the following expressions in L mol−1 s−1 units. k w1 = 3.3 × 103 × T 0.38 exp( −49.3 kJ mol−1/RT) k w2 = 1.7 × 107 × T −0.54 exp( −51.5 kJ mol−1/RT)

k w3 = 3.8 × 10−2 × T1.03 exp( −49.0 kJ mol−1/RT) k w4 = 4.0 × 10−12 × T 3.20 exp( −46.5 kJ mol−1/RT)

G V aG(s) = VMEP(s) + εint (s )

k4 ′ = 2.0 × 10−7 × T 4.26 exp( −46.4 kJ mol−1/RT)

where εGint(s) is the internal energy (rotational and vibrational) at s, which in the ground-state approximation equals the ZPE at s.

k5 = 1.7 × T 2.67 exp( −47.7 kJ mol−1/RT) G



Article

Table 5. Microcanonical Variational RRKM Results for Unimolecular Dissociation Reaction R8

a

E(ν,j)a

E#(ν,j)a

R#(Ȧ )a

E0a

N(ν,j)a × 1010

G(E#)a × 107

k(E) × 109(1/s)

299 320 341 362 383 404 467 529 655 739 885

16 38 61 83 105 128 194 260 392 480 633

3.41 3.29 3.25 3.21 3.18 3.16 3.11 3.06 2.99 2.95 2.89

283 282 280 279 278 276 273 269 263 259 252

0.008827 0.01615 0.02862 0.04928 0.08265 0.13540 0.52690 1.76200 14.0400 46.2500 284.700

0.000617 0.037190 0.530600 3.972000 20.28000 81.32000 2108.000 24260.00 899200.0 5726000 77950000

0.002096 0.06902 0.55580 2.41600 7.35700 18.010 119.90 412.60 1920.0 3711.0 8209.0

The same as those described in Table 4.

Table 6. Microcanonical Variational RRKM Results for Unimolecular Dissociation Reaction R14 E(ν,j)a E#(ν,j)a R#(Ȧ )a 293 418 586 628 732 753 a

21 146 433 478 585 636

3.40 3.39 2.29 2.28 2.27 2.20

E0a

N(ν,j)a × 103

G(E#)a × 103

k(E) × 1011(1/s)

272 272 153 150 147 117

0.1383 0.6451 2.924 4.009 8.167 9.309

0.2241 105.1 2189 3241 7551 12110

0.4858 48.82 224.4 242.4 277.2 390.0

Scheme 2


Table 7. Microcanonical Variational RRKM Results for Unimolecular Dissociation Reaction R16 E(ν,j)a E#(ν,j)a R#(Ȧ )a 386 394 419 436 452 469 511 578 a

36 45 154 175 194 213 258 327

3.54 3.53 3.15 3.14 3.13 3.12 3.11 3.10

E0a

N(ν,j)a × 104

G(E#)a × 105

k(E) × 1011(1/s)

350 349 265 261 258 256 253 251

0.6974 0.7686 1.018 1.217 1.446 1.709 2.534 4.490

0.1278 0.4479 5.251 8.136 11.93 16.79 34.76 87.22

0.5494 1.747 15.47 20.04 24.73 29.46 41.13 58.23


frequencies along the MEPs, the adiabatic ground-state potential VGa (s), the MEP potential VMEP(s), and ZPEs for channels R10, R12, R13, and R15 at the MPWB1K/631+G(d,p) level are shown in Figures 5−8. Our results imply that the variational effect is small for reactions R10, R12, R13, and R15. The variational calculations change the position of the saddle points by −0.015 to −0.021 Bohr. The variation of the normal-mode vibrational frequencies along the MEP for association channel R10 is shown in Figure 5. As shown in Figure 5, one of the stretching frequencies of S− H−O in van der Waals complex vdw1 changes from 2700 cm−1 in the reactants side at s = −3.5 (amu)1/2 Bohr to 1719 cm−1 in the saddle point at s = 0 (amu)1/2 Bohr and reaches a value of 3600 cm−1 on the product side at s = 3.5 (amu)1/2 Bohr that belongs to the O−H stretching mode in HO3. The same trend for the variation of major vibrations along the reaction coordinates for channels R12, R13, and R15 can be seen in Figures 6−8. To include the quantum effects in calculating the rate constants, the semiclassical ground-state probability PG(E) was calculated over the vibrational adiabatic potential by

The rate constants of reactions R10, R12, R13, and R15 (hydrogen abstraction channels) are calculated by using the dual-level direct dynamics approach. The MEP calculations are carried out by the IRC theory at the MPWB1K/6-31+G(d,p) level, and energetic information is further modified by the CCSD(T)/Aug-cc-pVTZ level. Dual-level dynamics calculations are carried out with the VTST-ISPE approach using interpolated single-point energies. Direct dynamics CVT calculations were carried out at the MPWB1K/6-31+G(d,p) level of theory for low-level calculations. The GAUSSRATE9.1 program, which is an interface between POLYRATE9.3.1 and Gaussian03, was employed in our CVT calculations. The Page− McIver algorithm was used to follow the MEP.51 The unharmonicity effect was not included in our CVT calculations. The normal-mode analysis was carried out in curvilinear coordinates at the transition state and in Cartesian coordinates at the minimum. A step size of 0.02 (amu)1/2 Bohr was used to calculate each individual point along the MEP, and Hessian calculation was performed at each 0.2 bohr (amu)1/2 Bohr interval. The variation of the normal-mode vibrational H



Article

Figure 4. Arrhenius plot of the calculated rate constants for various channels of the H2S + O3 reaction at 760 Torr pressure of N2. The solid line represents the result for reaction R9 and also the overall rate constant. The dashed−dotted line represents Rw3. The symbols are defined as (▲) Rw1, (Δ) Rw2, (■) Rw4, (□) R4′, (+) R5, (○) Rw6, (⬠) R7, (∇) R8, (◀) and (★) from refs 6 and 7, respectively, and (▶) from refs 48 and 49 for the loss of the reactants.

optimizing microcanonically (at every energy) the largest probability between the small curvature tunneling (SCT) probability,52 PSCT(E), and the large curvature tunneling (LCT) probability,53−56 PLCT(E), evaluated with the LCG4 version.57 The resulting probability is, therefore, given by ⎧ P SCT ⎫ ⎬ P μOMT(E) = max⎨ E ⎩ P LCT ⎭ ⎪

⎪

⎪

⎪

where μOMT stands for microcanonically optimized multidimensional tunneling.58 The rate constants for channels R10, R12, R13, and R15 were obtained using the transmission coefficient from the μOMT probability, κCVT/μOMT(T).59,60 Also, we calculated the thermally averaged transmission probability, P(E) exp(−E/RT), at several temperatures to examine the efficiency of zero curvature tunneling (ZCT) and SCT approximations. Our results showed that the probability values calculated from the SCT approximation are larger than those values calculated from the ZCT approximation. Curvilinear coordinates are used for vibrational frequency calculations around nonstationary points.61 In performing the dynamics calculations to obtain the rate constants for channels R10, R11, R12, R13, R14, R15, R16, and R16′, the geometries of the stationary points were optimized at the MPWB1K level of theory along with the 6-31+G(d,p) basis set. The IRC or MEP was constructed at the MPWB1K/631+G(d,p) level for low-level calculations and at the CCSD(T)/Aug-cc-pVTZ level for high-level corrections using the VTST-ISPE approach implemented in POLYRATE9.3.1. The Arrhenius plots for channels R10 and R11 are shown in Figure 9, and those for reactions R12, R13, R14, R15, R16, and R16′ are shown in Figure 10. The suggested rate constants by Goumri et al.62 for reactions R15 and R16′ are compared with our results in Figure 10. They theoretically studied the kinetics of some unimolecular reactions like reactions R15 and R16′ that occur in the H + SO2 system at the QCISD/6-311G(d,p) and G2 levels.

Figure 5. (a) Variation of frequencies along the MEP and (b) the vibrationally adiabatic ground-state potential VGa (s), MEP VMEP, and ZPE for reaction R10 at the MPWB1K level.

Two van der Waals complexes, vdw1 and vdw2 in Figure 2, were detected along the reaction coordinates R1 and R10 for vdw1 and R11 for vdw2. The IRC calculations that started at the saddle point geometry of TS1 go downhill toward the product valley to reach the Int1 and reactant valley to reach the van der Waals complex vdw1. The same procedure started at the saddle point geometry of TS11 to determine the structure and position of the van der Waals complex vdw2. As shown in Figure 2, no barrier was detected for the formation of the vdw1 complex, and its energy was found to be 2.5 kJ mol−1 lower than the total energy of the reactants at the CCSD(T) level. The vdw2 complex is formed on the product side of reaction R11 after passing over a high barrier height of 108.4 kJ mol−1. Releasing this much energy to the newly formed vdw2 complex makes this complex very unstable. Therefore, it has been assumed that the effect of formation of these two van der Waals complexes on the calculation of the rate constants is negligible. The calculated CVT/μOMT rate constant expressions for reactions R10−R16′ were found to be I



Article

Figure 6. (a) Variation of frequencies along the MEP and (b) the vibrationally adiabatic ground-state potential VGa (s), MEP VMEP, and ZPE for reaction R12 at the MPWB1K level. −1

k10 = 6.8 × 10

×T

2.3

Figure 7. (a)Variation of frequencies along the MEP and (b) the vibrationally adiabatic ground-state potential VGa (s), MEP VMEP, and ZPE for reaction R13 at the MPWB1K level.

−1

exp( −94.1 kJ mol /RT)

−1 −1

L mol s

k11 = 4.4 × 107 exp( −119.7 kJ mol−1/RT) L mol−1 s−1

multichannel RRKM-TST, and CVT methods on the lowest singlet surface. The suggested mechanism consists of 16 different channels with sulfur dioxide and water as the major products at lower temperatures. In calculating the rate constants for channels Rw1, Rw2, Rw3, Rw4, R4′, R5, Rw6, R7, R8, and R9, the effect of formation of four chemically activated intermediates plus energized sulfonic acid and sulfurous acid on the singlet surface are considered. It has been assumed that the lifetime of these chemically energized intermediates is long enough that the energy distribution in the activated species is statistical. It should be noticed that this kind of treatment to calculate the rate constants is based on a strong collision assumption that causes an overestimation of the rate of collisional stabilization of the intermediates. Our results indicate that this kind of treatment did not much affect the rate of the major channels (reactions R5, R7, R8, and R9) as the rates were much higher than that of the overestimated stabilization processes.

k12 = 5.4 × 104 × T 2.3 exp( −119.0 kJ mol−1/RT) s−1 k13 = 6.1 × 105 × T 2.1 exp( −184.5 kJ mol−1/RT) s−1 k14 = 4.3 × 1015 exp( −285.6 kJ mol−1/RT) s−1

k15 = 1.1 × 10−2 × T 4.1 exp( −212.5 kJ mol−1/RT) s−1 k16 = 3.6 × 1015 exp( −362.6 kJ mol−1/RT) s−1

■

k16 ′ = 3.6 × 1015 exp( −256.6 kJ mol−1/RT) s−1

CONCLUSION The kinetics and mechanism of the reaction of H2S + O3 is studied in detail by ab initio MO calculations and RRKM, J



Article

Figure 9. Arrhenius plot for reactions R10 and R11 at the CCSD(T)/ Aug-cc-pVTZ level.

Figure 8. (a)Variation of frequencies along the MEP and (b) the vibrationally adiabatic ground-state potential VGa (s), MEP VMEP, and ZPE for reaction R15 at the MPWB1K level.

Figure 10. Arrhenius plot at the CCSD(T)/Aug-cc-pVTZ level for channels R12−R16′. (□) and (■) are for channels R15 and R16′, respectively, at high pressure from ref 62.

■

CVT calculations were carried out to estimate the rate constants for the other channels (reactions R10−R16′). As shown in Figure 4, our method predicts that the overall rate at 300 K is in reasonable agreement with the values reported by refs 7, 48, and 49. At higher temperatures, formation of other products like O2, H2SO, and radicals like cis/trans-HOSO, SH, HO3, and OH also become important. The values of k15 and k16′ at 760 Torr in Figure 10 are in relatively good agreement with the values reported for these reactions at high pressure by Goumri et al.62 Association of H2S with triplet ozone forms energized intermediate Int5 that decomposes to H2SO + 3O2. In the previous studies on the kinetics of this system, the formation of sulfonic acid and sulfurous acid was ignored. The main reason for not observing these two species should be due to the lower potential barriers of TS5 and TS6 relative to TS3 and TS7 relative to TS6. The present study would be an initiator to study the dynamics of this system to explore its mechanism more accurately.

ASSOCIATED CONTENT

S Supporting Information *

The geometrical structures of all of the stationary points in the Z-matrix format are provided. This material is available free of charge via the Internet at http://pubs.acs.org.

■ ■ ■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS The financial support of the Research Council of Shiraz University is acknowledged. REFERENCES

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M


Multichannel RRKM-TST and Direct-Dynamics CVT Study of the

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