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A comparison of the stereo-dynamical information between S(1D) + H2 and S(1D) + HD Reactions
Computational and Theoretical Chemistry 1002 (2012) 9–15
Contents lists available at SciVerse ScienceDirect
Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
A comparison of the stereo-dynamical information between S(1D) + H2 and S(1D) + HD Reactions Yan Su a,⇑, Lihua Kang b a b
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China School of Chemistry and Chemical Engineering, Shihezi University, Shihezi 832003, China
a r t i c l e
i n f o
Article history: Received 1 March 2012 Received in revised form 11 September 2012 Accepted 13 September 2012 Available online 21 September 2012 Keywords: Quasi-classical trajectory Stereo-dynamics Reaction cross section
a b s t r a c t Studies on the stereo-dynamical of the reaction S(1D) + H2 and its isotopic variants have been performed using quasi-classical trajectory (QCT) method on a globally smooth ab initio potential surface of the 1A0 state which is produced by Ho et al. [11] at the collision energy of 0–22.0 kcal/mol. We present the reaction cross section as a function of collision energy, which show satisfactory agreement in the reaction cross section of Ho et al. [11]. Four polarization dependent generalized differential cross-sections (PDDCSs) have been calculated in the center-of-mass frame. Comparisons of the distributions P(hr), P(ur) and P(hr, ur), which denotes respectively the distribution of angles between k and j0 , the distribution of dihedral angle denoting k–k0 –j0 correlation and the angular distribution of product rotational vectors in the form of polar plots, are given for the isotopic variants and collision energies. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, significant progress has been made in atom–diatom insertion reactions [1–4], specifically, the S(1D) + H2 reaction and its isotopic variants have been widely investigated by theoretical and experimental methods, which play a central role in acid rain, air pollution, and global climate change [5]. A series of experiments have been carried out on the reaction S(1D) + H2 and its isotopic variants over the collision energies from 0.6 to 6 kcal/mol by Lee and Liu [6–8], where the vibration-state resolved differential cross-sections, and angle-resolved translational energy distributions were detailedly measured. From a theoretical point of view, Zyubin et al. [9] reported a new ab initio potential energy surface for the chemical reaction S(1D) + H2, where the quantum chemistry calculations were carried out at the multi-reference configuration interaction (MRCI) level with multi-configuration self-consistent field (MCSCF) reference wave functions. And the dynamics of the S(1D) + H2 reaction was simulated using the quasi-classical trajectory method. Simultaneously, Chao and Skodje [10] presented the results of a quasi-classical trajectory (QCT) calculation using a newly developed potential energy surface [11] for the title reactions, in which the integral cross sections, the product state-resolved differential cross sections, the angle-state-specific energy partitions, and other relevant reaction attributes were computed. Subsequently, a variety of quantum mechanical (QM) calculations have been performed on the reaction S(1D) + H2 and its isoto⇑ Corresponding author. E-mail address: [email protected] (Y. Su). 2210-271X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2012.09.008
pic variants. Honvault and Launay [12] presented the first accurate QM calculation for the S(1D) + H2 reaction at 2.24 kcal/mol collision energy on the latest PES by Ho et al. [11], where the results of accurate QM dynamical calculations are in agreement well with those of the experimental. Bañares et al. [13] have performed an accurate three-dimensional quantum mechanical (QM) scattering calculations of integral and differential cross sections. A time-dependent method is also used to calculate the state-to-state reaction probabilities for the S(1D) + H2 reaction by Mouret et al. [14]. Maiti et al. [15] have used a ‘‘mixed’’ representation approach in conjunction with a trajectory surface-hopping (TSH) method to study intersystem crossing effects in the S + H2 reaction based on high-quality potential surfaces. Lin and Guo [16] have subsequently performed quantum statistical and exact wave packet studies of the reactions between S(1D) and various hydrogen isotopes. Just recently, a nonadiabatic quantum dynamics calculation is reported for the S(1D) + HD reaction by Chu et al. [17] using the time-dependent wave packet approach. The results shown that the important discrepancy between theoretically calculated and experimentally measured intramolecular isotope effects can at least in part be attributed to significant nonadiabatic effects. Among the theoretical and experimental findings for S(1D) + H2 and its isotopic variants over the years, almost all the literature reports in this area are limited to the scalar properties, such as integral cross sections, differential cross sections, rate constants, reaction probability and branching ratio. The studies on the stereo-dynamics of S(1D) + H2 reaction are still rare, especially the isotopic effects on stereo-dynamic properties for the reactions S(1D) + H2 and S(1D) + HD. Hence, a detailed knowledge of the
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Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
thorough QCT method and explore the isotope effect on the product rotational polarization for two reactions in detail. 2. Computational methods 2.1. Rotational polarization of the product
Fig. 1. Comparison between the QCT-computed cross sections (the black curve) in this work and the previous TSH-computed cross section, single-surface reaction cross section (labeled single QCT), experimental data and Ho’s QCT reaction cross section. The blue curve is the experimental data from Ref. [8]. The red and magenta curves are the total TSH and single-surface reaction cross section from Ref. [15]. The olive curve is the Ho’s QCT reaction cross section. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
vector properties of the reactions is strongly desirable, which can provide the valuable information about chemical reaction stereo-dynamics. In this paper, we have selected S(1D) + H2 and S(1D) + HD reactions to force on their stereo-dynamics using a
It is known that the theory of the product rotational alignment has already been stated in numerous standard reports [18–26]. In the following, we will only give a simple description relevant to the present work. The center-of-mass (CM) frame is utilized in the calculations. The z-axis lies in the direction of the reagent relative velocity k, whereas the y-axis is perpendicular to the xz plane containing k and k0 . ht is the angle between the reagent relative velocity and product relative velocity (so-called scattering angle), corresponding to the angle between k and k0 . hr and ur are the polar and azimuthal angles of the final rotational angular momentum j0 . The distribution function P(hr) describing the k–k0 correlation can be expanded in a series of Legendre polynomials, and the dihedral angle distribution function P(ur) describing the k–k0 –j0 correlation can be expanded in a Fourier series. The full three-dimensional angular distribution associated with k–k0 –j0 can be represented by a set of generalized polarization dependent differential cross-sections (PDDCSs) in the CM frame that is described in Ref. [27]. 2.2. Potential energy surface and quasi-classical trajectory calculations All the theoretical calculations are carried out on the 1A0 H2S ab initio PES of Ho et al. [11] using the quasi-classical trajectory (QCT)
Fig. 2. The P(hr) distributions as a function of both the polar angle hr and the collision energy Ecol for the S(1D) + H2 and S(1D) + HD.
Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
method. The quasi-classical trajectory calculations by Han and coworkers developed code [18,21–23] are standard. Its efficiency and validity to investigate the product alignment of light light–light, heavy heavy–light, heavy light–light and light heavy–light mass combination reactions on attractive and repulsive potential energy surfaces have been approved in plenty of excellent work. Furthermore, as pointed out by previous studies [18–26], the QCT method can be of sufficiently high accuracy, which is in good with timeconsuming quantum mechanics for most chemical reactions. The QCT calculations have been performed over the collision energy range 0–22.0 kcal/mol. The vibrational v = 0 and rotational j = 0 levels of the reactant molecules are used throughout. The initial azimuthal orientation angle and polar angle of the reagent molecule inter-nuclear axis are randomly sampled using Monte Carlo method. In our calculations, 100,000 trajectories are sampled for each calculation and the integration step size is determined as 1017 s. The trajectories are started at an initial distance between the S atom and the center of mass of H2 or HD of 20.0 Å. 3. Results and discussion 3.1. Reaction cross section As emphasized above, reaction cross section as a function of collision energy is the key point to understand the dynamics of the title reaction. Fig. 1 compares the QCT calculated reaction cross section with previous trajectory surface-hopping (TSH) result and singlesurface reaction cross section (labeled single QCT) by Maiti et al. [15], the QCT reaction cross section of Ho et al. [11] and experimen-
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tal result of Lee and Liu [8]. The lowest singlet state PES determined by Ho et al. [11] has been employed in both the current QCT calculation and the previous calculations. It can be seen that the reaction cross section decreases with increasing collision energy. In Fig. 1, there is an obvious difference between our and Maiti’s single-surface reaction cross section, the difference resulted from that in Maiti’s report the potential energy surface was modified by adding 0.13626832898581 au globally, which will make the calculated reaction cross section smaller than that of the unmodified surface, as in our and Ho’s calculations [11]. Compared our result with Ho et al.’s [11], it is found that our calculated reaction cross section is in agreement well with Ho’s, as shown in Fig. 1, this indicates that our result is reliable and the QCT method has sufficiently high accuracy and efficiency to study the scalar and vector properties of the title reaction. 3.2. P(hr) distributions In a series of pioneering works, Han and co-workers reported that the distribution function P(hr) describing the k–j0 correlation is sensitive to two factors: the characters of PESs and the mass combination [19,21]. In this work, the detailed difference of P(hr) distributions on the different mass combination, whereas on the same potential energy surface is explored. Fig. 2a and b clearly illustrates that the peak of P(hr) distributions locate at hr = 90° and are symmetric with respect to 90°, whereas the span of curves is broad, which demonstrates that the product rotational angular momentum vector j0 is distributed with cylindrical symmetry in the product scattering frame and the direction of j0 is preferentially
Fig. 3. The P(ur) distributions as a function of both the polar angle ur and the collision energy Ecol (2.0, 6.0, 10.0, 14.0, 18.0, 22.0 kcal/mol, from inner to outer) for the S(1D) + H2 and S(1D) + HD.
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perpendicular to k direction. Comparing the P(hr) distributions for the reactions S(1D) + H2 and S(1D) + HD in Fig. 2 have indicated that the P(hr) distributions are strongly affected by mass combination, namely, notable isotope effects on stereo-dynamics. And specifically, the P(hr) distribution of the SH product for the S(1D) + H2 reaction in Fig. 2a and the P(hr) distribution of the SD product for the S(1D) + HD reaction in Fig. 2c have demonstrated that the increase of collision energy improve the intensity of product alignment, while the P(hr) distribution of the SH product for the S(1D) + HD reaction in Fig. 2b do not definitely expand monotonously with the increase of collision energy. However, the peaks at 90° in Fig. 2a and c are a little higher than that in Fig. 2b with the increase of the collision energy, indicating that the product rotational alignment against the direction perpendicular to k becomes stronger in the reactions S(1D) + H2 ? SH + H and S(1D) + HD ? SD + H. As shown in Fig. 2a and b, although the products for the first channel of the two reactions are the same (SH), the peak positions and outlines of the P(hr) distribution for the two reactions are obviously different. Fig. 2a and c shows that the P(hr) distribution are slightly different because the products for the second reaction channels of the two reactions are SH and SD. In the calculations, the PES of all reactions is the same. Thus, this difference of the P(hr) distributions is originated from the difference in the mass combination. According to cos2b = mAmC/ (mA + mB) (mB + mC) for the reaction A + BC ? AB + C and j0 = L sin2-
b + J cos2b + J1mB/mAB), the different mass mC will result in different values of j0 . Therefore, when the mC of H atom is substituted by the D atom, the P(hr) distribution describing the k, j0 correlation will be distinctly different. Emphatically, the P(hr) distribution displays very obvious isotopic effect for the reaction channels with the same product SH but weak effect for reaction channels with different products SH and SD. 3.3. P(ur) distributions The dihedral angular ion of the k–k0 –j0 three vectors correlation is characterized by the distribution function P(ur), which could provide some stereo-dynamical information on the alignment and orientation characters of the product as well. Fig. 3 shows the plots of P(ur) distributions for the reactions S(1D) + H2 and S(1D) + HD, respectively. Clearly, these P(ur) distributions tend to be asymmetric about ur = 180° with respect to the k–k0 scattering plane, which directly reflects the strong polarization of the product rotational angular momentum j0 . However, the rotation direction of the product is not isotropic. Two different peaks in the P(ur) distributions appearing around ur = 90° and ur = 270° imply that the rotational angular momentum vectors of product are preferentially aligned along the y-axis of the CM frame. With the increase of the collision energy, the peaks at ur = 90° in Fig. 3 are slightly stronger than that at ur = 270° for the reactions S(1D) + H2 ? SH + H,
Fig. 4. Polar plots of P(hr, ur) distributions averaged over all scattering angles at six different collision energies of 2.0, 6.0, 10.0, 14.0, 18.0, 22.0 kcal/mol for the S(1D) + H2 and S(1D) + HD.
Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
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Fig. 4. (continued)
S(1D) + HD ? SH + D and S(1D) + HD ? SD + H, which suggests that all the product rotational angular momentum vector j0 tend to be oriented along the positive direction of the y-axis. Therefore, although the H atom is substituted by the D atom, the orientation of the product SH from the first reaction channel and the product SH and SD from the second reaction channel for the two reactions will be the same, in which the P(ur) distribution of the first and second reaction channels are both come from the products SH. We could also observe that the alignment and orientation of the product molecules become stronger when the collision energy from 2.0 to 22.0 kcal/mol. Furthermore, the peaks of P(ur) distributions of the reactions S(1D) + H2 ? SH + H and S(1D) + HD ? SH + D become somewhat broader than the reaction S(1D) + HD ? SD + H, indicating that the rotation of the product molecule from S(1D) + HD ? SH + D has a preference of changing from the inplane reaction mechanism to the out-of-plane mechanism. 3.4. P(hr, ur) distributions The various angular distributions of the product that reflect the vector correlation contain rich information on the angular momentum polarization. To validate the information for the product molecular axis polarization, the P(hr, ur) distributions averaged over all scattering angles are plotted in Fig. 4. As presented in
Fig. 4, the P(hr, ur) distributions with the peaks at (90°, 90°), (90°, 270°) are in good accordance with the distributions of P(hr) and P(ur). The distributions of P(hr, ur) demonstrate that the products are strongly polarized perpendicular to the scattering plane and the products of reaction are mainly rotated in planes parallel to the scattering plane. 3.5. PDDCSs It is well known that polarization-dependent differential crosssection (PDDCS) contain information about the k–k0 –j0 correlation and the scattering direction of the product molecule. The four PDDCSs (2rp ddrx00t , 2rp ddrx20t , 2rp ddrx22þ and 2rp ddrx21 ) of products in the title t t reactions are displayed in Fig. 5. The PDDCS00 is simply the (k, k0 ) differential cross-section (DCS). It is obvious that all the calculated PDDCS00s both show a forward–backward asymmetry. Although all the scattering directions are changeless with increasing collision energy, the magnitude of PDDCS00 is different, in which the influence of scattering angle on PDDCS00 for S(1D) + HD ? SH + D is the largest. The PDDCS20 represents the expectation value of the second Legendre moment hP2 ðcos hÞi and it shows an opposite shape with PDDCS00, indicating that j0 is strongly aligned perpendicular to k at ht = 0° and ht = 180°. The values are close to 0.5 and 1.0 in averaged, which is a
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Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
Fig. 5. The four kinds of PDDCSs as a function of both the polar angle ht and the collision energy Ecol for the S(1D) + H2 and S(1D) + HD.
typical phenomenon for the insertion reaction and the PDDCS20s seem nearly isotropic scattering at other scattering angles. It can be seen from Fig. 5 that the negative value of PDDCS22+s reveal the product alignment along y-axis and the larger absolute value suggests the stronger alignment along the relevant axis. It is worth noting that alignment along the y axis for S(1D) + HD ? SH + H and S(1D) + HD ? SH + D will become stronger when the collision energy is increased, while that is opposite for the S(1D) + HD ? SD + H. Through comparison, we conclude that the above description is agree with the case of the P(ur) distribution functions. According to the negative value of PDDCS21s in Fig. 5, one could verdict that the products are aligned along the direction vector x + z at the whole scattering angle.
4. Conclusions The isotopic effects on stereo-dynamic properties for the reactions S(1D) + H2 and S(1D) + HD are investigated in detail by means of the quasi-classical trajectory method over the collision energy range 0–22.0 kcal/mol. We have calculated reaction cross section, the distributions of P(hr), P(ur), P(hr, ur) and PDDCSs of S(1D) + H2 and S(1D) + HD. The QCT-calculated reaction cross section is in good agreement with Ho et al.’s QCT result. The information of stereo-dynamic characters is sensitive to both energy and mass factor. The products for two reactions perpendicular to the initial velocity vector according to the P(hr) distributions, whereas the product SH of the S(1D) + HD display the weaker alignment. The distribution of
Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
P(ur) show that the rotational angular momentum vector of products is preferentially oriented along the positive direction of y-axis. The four normalized PDDCSs have been calculated and the results indicate that the reactions mainly both show forward and backward scattering. In addition, we find that the product rotational alignments have observably isotopic effects and will become slightly stronger with increasing collision energy. Acknowledgements This work is supported by the Fundamental Research Funds for Dalian University of Technology (No. DUT11RC(3)46) and the China Postdoctoral Science Foundation (No. 2012M510795). References [1] C. Piergiorgio, Chemical reaction dynamics with molecular beams, Rep. Prog. Phys. 63 (2000) 355–414. [2] K. Liu, Excitation functions of elementary chemical reactions: a direct link from crossed-beam dynamics to thermal kinetics, Int. Rev. Phys. Chem. 20 (2001) 189–217. [3] S.C. Althorpe, D.C. Clary, Quantum scattering calculations on chemical reactions, Ann. Rev. Phys. Chem. 54 (2003) 493–529. [4] E.J. Rackham, T. Gonzalez-Lezana, D.E. Manolopoulos, A rigorous test of the statistical model for atom–diatom insertion reactions, J. Chem. Phys. 119 (2003) 12895. [5] Y.Z. Song, P.J.S.B. Caridade, A.J.C. Varandas, Potential energy surface for groundstate H2S via scaling of the external correlation, comparison with extrapolation to complete basis set limit, and use in reaction dynamics, J. Phys. Chem. A 113 (2009) 9213–9219. [6] S.H. Lee, K. Liu, Direct mapping of insertion reaction dynamics: S(1D) + H2 ? SH + H, Appl. Phys. B: Lasers Opt. 71 (2000) 627–633. [7] S.-H. Lee, K. Liu, Exploring insertion reaction dynamics: a case study of S(1D) + D2 ? SD + D, J. Phys. Chem. A 102 (1998) 8637–8640. [8] S.-H. Lee, K. Liu, Isotope effects and excitation functions for the reactions of S(1D) + H2, D2 and HD, Chem. Phys. Lett. 290 (1998) 323–328. [9] A.S. Zyubin, A.M. Mebel, S.D. Chao, R.T. Skodje, Reaction dynamics of S(1D) + H2/D2 on a new ab initio potential surface, J. Chem. Phys. 114 (2001) 320. [10] S.D. Chao, R.T. Skodje, Quasi-classical trajectory studies of the insertion reactions S(1D) + H2, HD, and D2, J. Phys. Chem. A 105 (2001) 2474–2484.
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[11] T.S. Ho, T. Hollebeek, H. Rabitz, S.D. Chao, R.T. Skodje, A.S. Zyubin, A.M. Mebel, A globally smooth ab initio potential surface of the 1A0 state for the reaction S(1D) + H2, J. Chem. Phys. 116 (2002) 4124. [12] P. Honvault, J.M. Launay, Dynamics of the S(1D) + H2 ? SH + H reaction: a quantitative description using an accurate quantum method, Chem. Phys. Lett. 370 (2003) 371–375. [13] L. Bañares, F.J. Aoiz, P. Honvault, J.M. Launay, Dynamics of the S(1D) + H2 insertion reaction: a combined quantum mechanical and quasiclassical trajectory study, J. Phys. Chem. A 108 (2004) 1616–1628. [14] L. Mouret, J.-M. Launay, M. Terao-Dunseath, K. Dunseath, Time-dependent quantum dynamical study of C(1D) + H2 ? CH + H and S(1D) + H2?SH + H reactive collisions, Phys. Chem. Chem. Phys. 6 (2004) 4105–4110. [15] B. Maiti, G.C. Schatz, G. Lendvay, Importance of intersystem crossing in the S(3P, 1D) + H2 ? SH + H reaction, J. Phys. Chem. A 108 (2004) 8772–8781. [16] S.Y. Lin, H. Guo, Quantum statistical and wave packet studies of insertion reactions of S(1D) with H2, HD, and D2, J. Chem. Phys. 122 (2005) 074304. [17] T.S. Chu, K.L. Han, G.C. Schatz, Significant nonadiabatic effects in the S(1D) + HD reaction, J. Phys. Chem. A 111 (2007) 8286–8290. [18] K.L. Han, G.Z. He, N.Q. Lou, Effect of location of energy barrier on the product alignment of reaction A + BC, J. Chem. Phys. 105 (1996) 8699. [19] M.L. Wang, K.L. Han, G.Z. He, Product rotational polarization in photo-initiated bimolecular reactions A + BC: dependence on the character of the potential energy surface for different mass combinations, J. Phys. Chem. A 102 (1998) 10204–10210. [20] M.D. Chen, K.L. Han, N.Q. Lou, Theoretical study of stereodynamics for the reactions Cl + H2/HD/D2, J. Chem. Phys. 118 (2003) 4463. [21] M.L. Wang, K.L. Han, G.Z. He, Product rotational polarization in the photoinitiated bimolecular reaction A + BC ? AB + C on attractive, mixed and repulsive surfaces, J. Chem. Phys. 109 (1998) 5446. [22] M.D. Chen, K.L. Han, N.Q. Lou, Vector correlation in the H + D2 reaction and its isotopic variants: isotope effect on stereodynamics, Chem. Phys. Lett. 357 (2002) 483–490. [23] X. Zhang, K.L. Han, High-order symplectic integration in quasi-classical trajectory simulation: case study for O(1D) + H2, Int. J. Quantum Chem. 106 (2006) 1815–1819. [24] F.J. Aoiz, L. Banares, V.J. Herrero, Recent results from quasiclassical trajectory computations of elementary chemical reactions, J. Chem. Soc. Faraday Trans. 94 (1998) 2483–2500. [25] N.E. Shafer-Ray, A.J. Orr-Ewing, R.N. Zare, Beyond state-to-state differential cross sections: determination of product polarization in photoinitiated bimolecular reactions, J. Phys. Chem. 99 (1995) 7591–7603. [26] W.L. Li, M.S. Wang, C. Yang, W. Liu, C. Sun, T. Ren, Theoretical study of the stereodynamics of the reactions of D + H2 ? H + HD and H + D2 ? D + HD, Chem. Phys. 337 (2007) 93–98. [27] A.J. Orr-Ewing, R.N. Zare, Orientation and alignment of reaction products, Ann. Rev. Phys. Chem. 45 (1994) 315–366.
Contents lists available at SciVerse ScienceDirect
Computational and Theoretical Chemistry journal homepage: www.elsevier.com/locate/comptc
A comparison of the stereo-dynamical information between S(1D) + H2 and S(1D) + HD Reactions Yan Su a,⇑, Lihua Kang b a b
Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China School of Chemistry and Chemical Engineering, Shihezi University, Shihezi 832003, China
a r t i c l e
i n f o
Article history: Received 1 March 2012 Received in revised form 11 September 2012 Accepted 13 September 2012 Available online 21 September 2012 Keywords: Quasi-classical trajectory Stereo-dynamics Reaction cross section
a b s t r a c t Studies on the stereo-dynamical of the reaction S(1D) + H2 and its isotopic variants have been performed using quasi-classical trajectory (QCT) method on a globally smooth ab initio potential surface of the 1A0 state which is produced by Ho et al. [11] at the collision energy of 0–22.0 kcal/mol. We present the reaction cross section as a function of collision energy, which show satisfactory agreement in the reaction cross section of Ho et al. [11]. Four polarization dependent generalized differential cross-sections (PDDCSs) have been calculated in the center-of-mass frame. Comparisons of the distributions P(hr), P(ur) and P(hr, ur), which denotes respectively the distribution of angles between k and j0 , the distribution of dihedral angle denoting k–k0 –j0 correlation and the angular distribution of product rotational vectors in the form of polar plots, are given for the isotopic variants and collision energies. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, significant progress has been made in atom–diatom insertion reactions [1–4], specifically, the S(1D) + H2 reaction and its isotopic variants have been widely investigated by theoretical and experimental methods, which play a central role in acid rain, air pollution, and global climate change [5]. A series of experiments have been carried out on the reaction S(1D) + H2 and its isotopic variants over the collision energies from 0.6 to 6 kcal/mol by Lee and Liu [6–8], where the vibration-state resolved differential cross-sections, and angle-resolved translational energy distributions were detailedly measured. From a theoretical point of view, Zyubin et al. [9] reported a new ab initio potential energy surface for the chemical reaction S(1D) + H2, where the quantum chemistry calculations were carried out at the multi-reference configuration interaction (MRCI) level with multi-configuration self-consistent field (MCSCF) reference wave functions. And the dynamics of the S(1D) + H2 reaction was simulated using the quasi-classical trajectory method. Simultaneously, Chao and Skodje [10] presented the results of a quasi-classical trajectory (QCT) calculation using a newly developed potential energy surface [11] for the title reactions, in which the integral cross sections, the product state-resolved differential cross sections, the angle-state-specific energy partitions, and other relevant reaction attributes were computed. Subsequently, a variety of quantum mechanical (QM) calculations have been performed on the reaction S(1D) + H2 and its isoto⇑ Corresponding author. E-mail address: [email protected] (Y. Su). 2210-271X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comptc.2012.09.008
pic variants. Honvault and Launay [12] presented the first accurate QM calculation for the S(1D) + H2 reaction at 2.24 kcal/mol collision energy on the latest PES by Ho et al. [11], where the results of accurate QM dynamical calculations are in agreement well with those of the experimental. Bañares et al. [13] have performed an accurate three-dimensional quantum mechanical (QM) scattering calculations of integral and differential cross sections. A time-dependent method is also used to calculate the state-to-state reaction probabilities for the S(1D) + H2 reaction by Mouret et al. [14]. Maiti et al. [15] have used a ‘‘mixed’’ representation approach in conjunction with a trajectory surface-hopping (TSH) method to study intersystem crossing effects in the S + H2 reaction based on high-quality potential surfaces. Lin and Guo [16] have subsequently performed quantum statistical and exact wave packet studies of the reactions between S(1D) and various hydrogen isotopes. Just recently, a nonadiabatic quantum dynamics calculation is reported for the S(1D) + HD reaction by Chu et al. [17] using the time-dependent wave packet approach. The results shown that the important discrepancy between theoretically calculated and experimentally measured intramolecular isotope effects can at least in part be attributed to significant nonadiabatic effects. Among the theoretical and experimental findings for S(1D) + H2 and its isotopic variants over the years, almost all the literature reports in this area are limited to the scalar properties, such as integral cross sections, differential cross sections, rate constants, reaction probability and branching ratio. The studies on the stereo-dynamics of S(1D) + H2 reaction are still rare, especially the isotopic effects on stereo-dynamic properties for the reactions S(1D) + H2 and S(1D) + HD. Hence, a detailed knowledge of the
10
Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
thorough QCT method and explore the isotope effect on the product rotational polarization for two reactions in detail. 2. Computational methods 2.1. Rotational polarization of the product
Fig. 1. Comparison between the QCT-computed cross sections (the black curve) in this work and the previous TSH-computed cross section, single-surface reaction cross section (labeled single QCT), experimental data and Ho’s QCT reaction cross section. The blue curve is the experimental data from Ref. [8]. The red and magenta curves are the total TSH and single-surface reaction cross section from Ref. [15]. The olive curve is the Ho’s QCT reaction cross section. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
vector properties of the reactions is strongly desirable, which can provide the valuable information about chemical reaction stereo-dynamics. In this paper, we have selected S(1D) + H2 and S(1D) + HD reactions to force on their stereo-dynamics using a
It is known that the theory of the product rotational alignment has already been stated in numerous standard reports [18–26]. In the following, we will only give a simple description relevant to the present work. The center-of-mass (CM) frame is utilized in the calculations. The z-axis lies in the direction of the reagent relative velocity k, whereas the y-axis is perpendicular to the xz plane containing k and k0 . ht is the angle between the reagent relative velocity and product relative velocity (so-called scattering angle), corresponding to the angle between k and k0 . hr and ur are the polar and azimuthal angles of the final rotational angular momentum j0 . The distribution function P(hr) describing the k–k0 correlation can be expanded in a series of Legendre polynomials, and the dihedral angle distribution function P(ur) describing the k–k0 –j0 correlation can be expanded in a Fourier series. The full three-dimensional angular distribution associated with k–k0 –j0 can be represented by a set of generalized polarization dependent differential cross-sections (PDDCSs) in the CM frame that is described in Ref. [27]. 2.2. Potential energy surface and quasi-classical trajectory calculations All the theoretical calculations are carried out on the 1A0 H2S ab initio PES of Ho et al. [11] using the quasi-classical trajectory (QCT)
Fig. 2. The P(hr) distributions as a function of both the polar angle hr and the collision energy Ecol for the S(1D) + H2 and S(1D) + HD.
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method. The quasi-classical trajectory calculations by Han and coworkers developed code [18,21–23] are standard. Its efficiency and validity to investigate the product alignment of light light–light, heavy heavy–light, heavy light–light and light heavy–light mass combination reactions on attractive and repulsive potential energy surfaces have been approved in plenty of excellent work. Furthermore, as pointed out by previous studies [18–26], the QCT method can be of sufficiently high accuracy, which is in good with timeconsuming quantum mechanics for most chemical reactions. The QCT calculations have been performed over the collision energy range 0–22.0 kcal/mol. The vibrational v = 0 and rotational j = 0 levels of the reactant molecules are used throughout. The initial azimuthal orientation angle and polar angle of the reagent molecule inter-nuclear axis are randomly sampled using Monte Carlo method. In our calculations, 100,000 trajectories are sampled for each calculation and the integration step size is determined as 1017 s. The trajectories are started at an initial distance between the S atom and the center of mass of H2 or HD of 20.0 Å. 3. Results and discussion 3.1. Reaction cross section As emphasized above, reaction cross section as a function of collision energy is the key point to understand the dynamics of the title reaction. Fig. 1 compares the QCT calculated reaction cross section with previous trajectory surface-hopping (TSH) result and singlesurface reaction cross section (labeled single QCT) by Maiti et al. [15], the QCT reaction cross section of Ho et al. [11] and experimen-
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tal result of Lee and Liu [8]. The lowest singlet state PES determined by Ho et al. [11] has been employed in both the current QCT calculation and the previous calculations. It can be seen that the reaction cross section decreases with increasing collision energy. In Fig. 1, there is an obvious difference between our and Maiti’s single-surface reaction cross section, the difference resulted from that in Maiti’s report the potential energy surface was modified by adding 0.13626832898581 au globally, which will make the calculated reaction cross section smaller than that of the unmodified surface, as in our and Ho’s calculations [11]. Compared our result with Ho et al.’s [11], it is found that our calculated reaction cross section is in agreement well with Ho’s, as shown in Fig. 1, this indicates that our result is reliable and the QCT method has sufficiently high accuracy and efficiency to study the scalar and vector properties of the title reaction. 3.2. P(hr) distributions In a series of pioneering works, Han and co-workers reported that the distribution function P(hr) describing the k–j0 correlation is sensitive to two factors: the characters of PESs and the mass combination [19,21]. In this work, the detailed difference of P(hr) distributions on the different mass combination, whereas on the same potential energy surface is explored. Fig. 2a and b clearly illustrates that the peak of P(hr) distributions locate at hr = 90° and are symmetric with respect to 90°, whereas the span of curves is broad, which demonstrates that the product rotational angular momentum vector j0 is distributed with cylindrical symmetry in the product scattering frame and the direction of j0 is preferentially
Fig. 3. The P(ur) distributions as a function of both the polar angle ur and the collision energy Ecol (2.0, 6.0, 10.0, 14.0, 18.0, 22.0 kcal/mol, from inner to outer) for the S(1D) + H2 and S(1D) + HD.
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perpendicular to k direction. Comparing the P(hr) distributions for the reactions S(1D) + H2 and S(1D) + HD in Fig. 2 have indicated that the P(hr) distributions are strongly affected by mass combination, namely, notable isotope effects on stereo-dynamics. And specifically, the P(hr) distribution of the SH product for the S(1D) + H2 reaction in Fig. 2a and the P(hr) distribution of the SD product for the S(1D) + HD reaction in Fig. 2c have demonstrated that the increase of collision energy improve the intensity of product alignment, while the P(hr) distribution of the SH product for the S(1D) + HD reaction in Fig. 2b do not definitely expand monotonously with the increase of collision energy. However, the peaks at 90° in Fig. 2a and c are a little higher than that in Fig. 2b with the increase of the collision energy, indicating that the product rotational alignment against the direction perpendicular to k becomes stronger in the reactions S(1D) + H2 ? SH + H and S(1D) + HD ? SD + H. As shown in Fig. 2a and b, although the products for the first channel of the two reactions are the same (SH), the peak positions and outlines of the P(hr) distribution for the two reactions are obviously different. Fig. 2a and c shows that the P(hr) distribution are slightly different because the products for the second reaction channels of the two reactions are SH and SD. In the calculations, the PES of all reactions is the same. Thus, this difference of the P(hr) distributions is originated from the difference in the mass combination. According to cos2b = mAmC/ (mA + mB) (mB + mC) for the reaction A + BC ? AB + C and j0 = L sin2-
b + J cos2b + J1mB/mAB), the different mass mC will result in different values of j0 . Therefore, when the mC of H atom is substituted by the D atom, the P(hr) distribution describing the k, j0 correlation will be distinctly different. Emphatically, the P(hr) distribution displays very obvious isotopic effect for the reaction channels with the same product SH but weak effect for reaction channels with different products SH and SD. 3.3. P(ur) distributions The dihedral angular ion of the k–k0 –j0 three vectors correlation is characterized by the distribution function P(ur), which could provide some stereo-dynamical information on the alignment and orientation characters of the product as well. Fig. 3 shows the plots of P(ur) distributions for the reactions S(1D) + H2 and S(1D) + HD, respectively. Clearly, these P(ur) distributions tend to be asymmetric about ur = 180° with respect to the k–k0 scattering plane, which directly reflects the strong polarization of the product rotational angular momentum j0 . However, the rotation direction of the product is not isotropic. Two different peaks in the P(ur) distributions appearing around ur = 90° and ur = 270° imply that the rotational angular momentum vectors of product are preferentially aligned along the y-axis of the CM frame. With the increase of the collision energy, the peaks at ur = 90° in Fig. 3 are slightly stronger than that at ur = 270° for the reactions S(1D) + H2 ? SH + H,
Fig. 4. Polar plots of P(hr, ur) distributions averaged over all scattering angles at six different collision energies of 2.0, 6.0, 10.0, 14.0, 18.0, 22.0 kcal/mol for the S(1D) + H2 and S(1D) + HD.
Y. Su, L. Kang / Computational and Theoretical Chemistry 1002 (2012) 9–15
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Fig. 4. (continued)
S(1D) + HD ? SH + D and S(1D) + HD ? SD + H, which suggests that all the product rotational angular momentum vector j0 tend to be oriented along the positive direction of the y-axis. Therefore, although the H atom is substituted by the D atom, the orientation of the product SH from the first reaction channel and the product SH and SD from the second reaction channel for the two reactions will be the same, in which the P(ur) distribution of the first and second reaction channels are both come from the products SH. We could also observe that the alignment and orientation of the product molecules become stronger when the collision energy from 2.0 to 22.0 kcal/mol. Furthermore, the peaks of P(ur) distributions of the reactions S(1D) + H2 ? SH + H and S(1D) + HD ? SH + D become somewhat broader than the reaction S(1D) + HD ? SD + H, indicating that the rotation of the product molecule from S(1D) + HD ? SH + D has a preference of changing from the inplane reaction mechanism to the out-of-plane mechanism. 3.4. P(hr, ur) distributions The various angular distributions of the product that reflect the vector correlation contain rich information on the angular momentum polarization. To validate the information for the product molecular axis polarization, the P(hr, ur) distributions averaged over all scattering angles are plotted in Fig. 4. As presented in
Fig. 4, the P(hr, ur) distributions with the peaks at (90°, 90°), (90°, 270°) are in good accordance with the distributions of P(hr) and P(ur). The distributions of P(hr, ur) demonstrate that the products are strongly polarized perpendicular to the scattering plane and the products of reaction are mainly rotated in planes parallel to the scattering plane. 3.5. PDDCSs It is well known that polarization-dependent differential crosssection (PDDCS) contain information about the k–k0 –j0 correlation and the scattering direction of the product molecule. The four PDDCSs (2rp ddrx00t , 2rp ddrx20t , 2rp ddrx22þ and 2rp ddrx21 ) of products in the title t t reactions are displayed in Fig. 5. The PDDCS00 is simply the (k, k0 ) differential cross-section (DCS). It is obvious that all the calculated PDDCS00s both show a forward–backward asymmetry. Although all the scattering directions are changeless with increasing collision energy, the magnitude of PDDCS00 is different, in which the influence of scattering angle on PDDCS00 for S(1D) + HD ? SH + D is the largest. The PDDCS20 represents the expectation value of the second Legendre moment hP2 ðcos hÞi and it shows an opposite shape with PDDCS00, indicating that j0 is strongly aligned perpendicular to k at ht = 0° and ht = 180°. The values are close to 0.5 and 1.0 in averaged, which is a
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Fig. 5. The four kinds of PDDCSs as a function of both the polar angle ht and the collision energy Ecol for the S(1D) + H2 and S(1D) + HD.
typical phenomenon for the insertion reaction and the PDDCS20s seem nearly isotropic scattering at other scattering angles. It can be seen from Fig. 5 that the negative value of PDDCS22+s reveal the product alignment along y-axis and the larger absolute value suggests the stronger alignment along the relevant axis. It is worth noting that alignment along the y axis for S(1D) + HD ? SH + H and S(1D) + HD ? SH + D will become stronger when the collision energy is increased, while that is opposite for the S(1D) + HD ? SD + H. Through comparison, we conclude that the above description is agree with the case of the P(ur) distribution functions. According to the negative value of PDDCS21s in Fig. 5, one could verdict that the products are aligned along the direction vector x + z at the whole scattering angle.
4. Conclusions The isotopic effects on stereo-dynamic properties for the reactions S(1D) + H2 and S(1D) + HD are investigated in detail by means of the quasi-classical trajectory method over the collision energy range 0–22.0 kcal/mol. We have calculated reaction cross section, the distributions of P(hr), P(ur), P(hr, ur) and PDDCSs of S(1D) + H2 and S(1D) + HD. The QCT-calculated reaction cross section is in good agreement with Ho et al.’s QCT result. The information of stereo-dynamic characters is sensitive to both energy and mass factor. The products for two reactions perpendicular to the initial velocity vector according to the P(hr) distributions, whereas the product SH of the S(1D) + HD display the weaker alignment. The distribution of
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P(ur) show that the rotational angular momentum vector of products is preferentially oriented along the positive direction of y-axis. The four normalized PDDCSs have been calculated and the results indicate that the reactions mainly both show forward and backward scattering. In addition, we find that the product rotational alignments have observably isotopic effects and will become slightly stronger with increasing collision energy. Acknowledgements This work is supported by the Fundamental Research Funds for Dalian University of Technology (No. DUT11RC(3)46) and the China Postdoctoral Science Foundation (No. 2012M510795). References [1] C. Piergiorgio, Chemical reaction dynamics with molecular beams, Rep. Prog. Phys. 63 (2000) 355–414. [2] K. Liu, Excitation functions of elementary chemical reactions: a direct link from crossed-beam dynamics to thermal kinetics, Int. Rev. Phys. Chem. 20 (2001) 189–217. [3] S.C. Althorpe, D.C. Clary, Quantum scattering calculations on chemical reactions, Ann. Rev. Phys. Chem. 54 (2003) 493–529. [4] E.J. Rackham, T. Gonzalez-Lezana, D.E. Manolopoulos, A rigorous test of the statistical model for atom–diatom insertion reactions, J. Chem. Phys. 119 (2003) 12895. [5] Y.Z. Song, P.J.S.B. Caridade, A.J.C. Varandas, Potential energy surface for groundstate H2S via scaling of the external correlation, comparison with extrapolation to complete basis set limit, and use in reaction dynamics, J. Phys. Chem. A 113 (2009) 9213–9219. [6] S.H. Lee, K. Liu, Direct mapping of insertion reaction dynamics: S(1D) + H2 ? SH + H, Appl. Phys. B: Lasers Opt. 71 (2000) 627–633. [7] S.-H. Lee, K. Liu, Exploring insertion reaction dynamics: a case study of S(1D) + D2 ? SD + D, J. Phys. Chem. A 102 (1998) 8637–8640. [8] S.-H. Lee, K. Liu, Isotope effects and excitation functions for the reactions of S(1D) + H2, D2 and HD, Chem. Phys. Lett. 290 (1998) 323–328. [9] A.S. Zyubin, A.M. Mebel, S.D. Chao, R.T. Skodje, Reaction dynamics of S(1D) + H2/D2 on a new ab initio potential surface, J. Chem. Phys. 114 (2001) 320. [10] S.D. Chao, R.T. Skodje, Quasi-classical trajectory studies of the insertion reactions S(1D) + H2, HD, and D2, J. Phys. Chem. A 105 (2001) 2474–2484.
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