Hongwei Song and Hua Guo * Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico, 87131, USA

Transcription

1 Submitted to JCP, 10/21/2014 Effects of reactant rotational excitations on + N H + NH 3 reactivity Hongwei Song and Hua Guo * Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico, 87131, USA * : Corresponding author, hguo@unm.edu 1

2 Abstract Rotational mode specificity of the title reaction is examined using an initial state selected time-dependent wave packet method on an accurate ab initio based global potential energy surface. This penta-atomic reaction presents an ideal system to test several dynamical approximations, which might be useful for future quantum dynamics studies of polyatomic reactions, particularly with rotationally excited reactants. The first approximation involves a seven-dimensional (7D) model in which the two non-reactive N-H bonds are fixed at their equilibrium geometry. The second is the centrifugal sudden approximation within the 7D model. Finally, the J-shifting model is tested, again with the fixed N-H bonds. The spectator-bond approximation works very well in the energy range studied, while the centrifugal sudden and J-shifting integral cross sections agree satisfactorily with the coupled-channel counterparts in the low collision energy range, but deviate at the high energies. The calculated integral cross sections indicate that the rotational excitation of somewhat inhibits the reaction while the rotational excitations of N have little effect. These findings are compared with the predictions of the Sudden Vector Proection model. Finally, a simple model is proposed to predict rotational mode specificity using K-averaged reaction probabilities. 2

3 I. INTRODUCTION A complete understanding of chemical reaction dynamics requires a quantum mechanical treatment. 1-3 Thus far, most quantum dynamical studies of bimolecular reactions have focused on rotationless reactants. This is partly because very few experimental studies of reaction dynamics have been performed with rotationally excited reactants, due apparently to the difficulties associated with the preparation of a single rotational state for the reactant. 4-6 In addition, a rigorous quantum dynamical treatment of a reaction involving rotationally excited reactants is also computationally costly. If a wave packet method is used, for example, the dynamics of a reaction with 0 would require roughly 2 1 wave packets, as opposed to one wave packet i i for i 0. 7 Nonetheless, the reactant rotational degrees of freedom have been shown to play an important role in dynamics of some reactions. 4-6, 8 As a result, it is highly desirable to understand the rotational mode specificity, particularly for reactions with molecular reactants. Indeed, several recent quantum dynamics studies have been 6, 9-22 devoted to the exploration of rotational effects in bimolecular reactions. In this work, we focus on the rotational mode specificity in a prototypical penta-atom reactive system, namely the + N H + NH 3 reaction. This reaction has been the subect of extensive theoretical investigations It provides an ideal proving ground for studying mode specificity because both reactants are molecules and their rotational degrees of freedom can conceivably affect the reaction dynamics. 3

4 Technically, an accurate global potential energy surface (PES) has recently been developed by fitting ~100,000 high-level ab initio points, 33 using a high fidelity permutation invariant polynomial-neural network (PIP-NN) method Furthermore, full dimensional quantum dynamics calculations on the accurate PIP-NN PES have been reported, 36 providing an important benchmark for more approximate methods tested in this work. The last point is particularly important, as the computational costs for larger reactive systems such as the current system are formidable, and as a result, dynamical approximations are of great value in predicting trends in mode specificity. In this work, an initial state selected time-dependent wave packet method is employed to study the + N reaction within a reduced seven-dimensional model by fixing the two NH bonds in the reactant N. Since the lengths of the two NH bonds in N are almost unchanged along the minimum energy path, the reduced-dimensional model seems to be appropriate, at least in the low energy region interested in this study. We further test the accuracy of the centrifugal sudden (CS) approximation and J-shifting (JS) approximation and use them to study the effects of rotational excitations of both reactants on the title reaction. The observed rotational effects are compared with the predictions of the recently proposed Sudden Vector Proection model Finally, a simple model is proposed to predict rotational mode specificity using helicity-averaged reaction probabilities. This paper is organized as follows. Section II outlines the theoretical methodology of the initial state selected 4

5 wave packet method. The results and discussion are presented in Section III. We conclude in Section IV. II. THEORY By fixing the two non-reactive bonds, the seven-dimensional Hamiltonian for the penta-atomic reaction AB + CDE in the reactant Jacobi coordinates, as shown in Fig. 1, for a given total angular momentum quantum number ( J tot ) can be written as ( 1 hereafter): ( ˆ ˆ) ˆ ˆ ˆ ˆ ˆ J J l H ( ) 2 R 2 R 2 r 2 r 2 r R tot h 2 1 r R Vˆ( R, r, r, r,,,,, ) V ( r ), ref (1) where R is the distance between the centers of mass (COMs) of AB and E-CD, r 1 is the bond distance of AB, r 3 is the bond distance of CD and r 2 is the distance from E to the COM of CD, which are fixed at r 30 = a 0 and r 20 = a 0, respectively. Since the N atom (D) is much heavier than the H atom (C and E), this is approximately equivalent to fixing the two NH bonds. The corresponding reduced masses are given by R, 1, 2, and 3. ĵ 1 is the rotational angular momentum operator of AB. ĵ 3 is the rotational angular momentum operator of CD and ˆl 2 is the orbital angular momentum operator of atom E with respect to CD. ĵ 23 is the angular momentum operator of CDE, which is the sum of ĵ 3 and ˆl 2. J ˆtot is the 5

6 total angular momentum operator of the system and Ĵ is the sum of ĵ and 1 ĵ 23. h ˆ 1 ( r 1 ) is the one-dimensional (1D) reference Hamiltonian, which is defined as ˆ 1 h ( r ) V ( r ), ref 2 21 r1 1 (2) ref where V ( r 1) is the corresponding 1D reference potential. The parity () adapted wave function is expanded in terms of the body-fixed (BF) rovibrational basis functions: ( R, r, r, r ) F u ( R) ( r ) ( Rˆ, rˆ, rˆ, rˆ ), (3) JtotM JtotM v1 JtotM n1k n 1 1 K n, 1,, K where n labels the translational basis functions, 1 is the basis index for r, and 1 1 denotes ( 1, l2, 3, 23, J ). The translational basis function, u, is dependent on 1 v n due to the use of an L-shaped grid. 41 Jtot M in Eq. (3) is the parity-adapted coupled K BF total angular momentum eigenfunction, which can be written as Jtot M K 2 2J 1 ( 1 ) 1 tot Jtot * JK 1 l2 3J Jtot Jtot * J K K 0 [ DK, MY ( 1) ], 1 23l2 D 3 K, MY (4) 1 23l2 3 8 J where D, is the Wigner rotation matrix. 42 M is the proection of J tot on the KM space-fixed z axis, and K is its proection on the BF z axis that coincides with R. Y is the eigenfunction of Ĵ defined as JK l JK Y ˆ ˆ ˆ 1 23l K JK y ( 1 r1 ) Y ( 23l2 3K r2, r3 ), (5) and 6

7 2l 1 Y ( rˆ, rˆ ) D ( rˆ ) ml 0 m y ( rˆ ), (6) l2 3K 2 3 K m m 3 m where ω is the proection of 1 on the BF z axis and y m denotes the spherical 1 l2 3 J J harmonics. Note the restriction that ( 1) tot 1 for K 0 in Eq. (4). The centrifugal term, i.e. ( Jˆ ˆ) 2 tot J in the Hamiltonian, which gives rise to the coupling between different K blocks in the BF representation, is given by Jtot M K ( Jˆ tot Jˆ) 2 Jtot M K { K 1K K 1K KK [ J Jtot K Jtot K tot ( J (1 JK (1 JK tot 1) J ( J 1) 2K ) 1/ 2 K 0 ) } 1/ 2 K1 2 ] (7) 1/ 2 with [ A( A 1) B( B 1)]. Within the CS approximation, the couplings AB between different K blocks are neglected, making K a good quantum number. The initial wave packet is chosen as the direct product of a localized Gaussian wave packet in the scattering coordinate (R) and a specific ( JtotK ) state of the reactive system with specific internal rovibrational states of both and N. The latter were obtained by diagonalizing the individual rovibrational Hamiltonians. This wave packet is then propagated in time using the second order split-operator method. 45 F The flux through the dividing surface, S[ r1 r1 ], is obtained from the energy-dependent scattering wavefunction, ( E), which is determined by Fourier transforming the wave packet at the dividing surface. Finally, the wave packet is absorbed at the edges of the grid using an absorbing potential with the form given in i Table I. 7

8 The integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all the partial waves (total angular momentum J tot ), 1 ( E) (2J 1) P ( E) Jtot vh vnh 2 2 H 2 NH2 2 tot vh v 2 NH 2 H 2 NH JK 2 (2 H 1)(2 2 NH 1) JK k J 2 tot K 1 (2 1)(2 1) H2 NH2 JK JK vh v 2 NH 2 H 2 NH2 ( E), (8) where is defined as the J, K and ԑ specific cross section and K is taken JK vh v 2 NH 2 H 2 NH2 from 0 to min(j, J tot ). is the initial rotational quantum number of and N represents the initial rotational state of the N asymmetric rotor labeled by J. In this study, we focus on the rotational mode specificity with the reactants in their ground vibrational states, i.e., v 0, v (0,0,0). Thus, we will drop the H2 NH2 vibrational quantum numbers hereafter. III. RESULTS The numerical parameters employed in the calculations on an L-shaped grid are given in Table I. For the scattering coordinate R, 70 sinc discrete variable representation 46 (DVR) bases/points are used in the full range from 1.5 to 11.5 a 0 and 27 sine DVR bases/points are used in the interaction region. For the reactive H-H distance of r 1, 21 potential optimized DVR basis/points are used in the interaction region, and 3 PODVR basis/points are used in the asymptotic region. The large angular basis is controlled by 1max, 23max, 3max, l 2max, and J max and has a dimension of 8

9 around for K=0 and for K>0. The evaluation of the potential energy operator is efficiently carried out by a pseudo-spectral method. 46 The propagation time is around 6500 a.u. with a time step of 10 a.u. A. Accuracy of approximations Very recently, we carried out a full-dimensional quantum dynamical study of the title reaction under the CS approximation. 36 In order to test the accuracy of the reduced seven-dimensional model (i.e. the spectator-bond approximation) employed in the current work, we first calculate the ICS for reactants both in their ground rovibrational states under the CS approximation and compare the corresponding excitation function with the full-dimensional CS result in Fig. 2(a). As can be seen from this figure, the 7D ICS agrees well with the 9D result. They are nearly indistinguishable over the entire energy range, although the 7D cross sections appear to be slightly larger than the corresponding 9D results at high collision energies. The agreement indicates that the two NH bonds can indeed be treated as spectators for the title reaction, at least in the energy range studied here. Based on the fact that the 7D model works very well for this reaction, we further test the accuracy of the CS approximation by comparing its cross section with the coupled-channel (CC) result. The 7D CC cross section from the ground rovibrational states of the two reactants is also presented in Fig. 2(a). Clearly, the CS cross section resembles the CC result well in the low collision energy range and it becomes visibly 9

10 larger than the CC result for collision energies higher than 0.65 ev. The relative error is less than 15%. Overall, the CC excitation function is well reproduced by the CS calculation. An even simpler and less accurate model is the JS approximation, 47 in which the reaction probabilities for J tot > K are calculated by shifting the collision energy in the Jtot K Jtot K probability for J tot =K, i.e. P ( E) P ( E E) with E B * [ J ( J 1) K( K 1)]. 48 The rotational constant B * is taken as cm -1, tot tot which is obtained from the geometry of the saddle point of the PES. The JS cross section from the ground rovibrational states of the reactants is given in Fig. 2(a). It is clear from the figure that the JS approximation works well at low collision energies while it significantly overestimates the ICS at high collision energies with the relative error up to 30 %. The good agreement in the low energy range can be readily explained by the fact that this reaction has an early barrier with a height of ev, which presumably has weak coupling between the vibrational and rotational coordinates. Since the CS approximation apparently provides a reasonable compromise between accuracy and computational costs, it is chosen as the method to generate the results presented below unless stated otherwise. B. Rotational effects of and N 10

11 N ( J Figure 3(a) shows the K-specific and K-averaged ICSs for the ( =1) + = 0 00 ) reaction as a function of the translational energy. It can be seen that the ICS decreases and the energy threshold shifts to a slightly higher energy with the increasing K. The K-averaged cross section is between those for K = 0 and K = 1 and close to that for K = 1 because of its larger (2 ) weight. The K-specific and K-averaged ICSs for = 2 are presented in Fig. 3(b). Similar to the = 1 case, the ICS decreases with the increasing K and the reaction energy threshold raises. Thus, the helicity quantum number, K, has a significant impact on the reaction. It is generally recognized that the effect of low rotational excitation manifests itself as an orientation effect while the highly excited states influence the reactivity as an energy effect. 9 The significant effect here is almost certainly caused by the different initial orientations of the reactants resulting from the different values of K. In Fig. 3(c), we present the K-averaged ICSs for = 0, 1 and 2 as a function of the translational energy. As increases, the ICS decreases. The reaction energy threshold is, however, almost unchanged. In other words, the initial rotational excitation of reduces the magnitude of the ICS but has little effect on the energy threshold. The K-specific and K-averaged ICSs for the ( =0) + N ( J =1 11 ) reaction are plotted in Fig. 4(a) as a function of the translational energy. The cross section for K = 1 is much larger than that for K = 0 and the reaction energy threshold is shifted to a lower energy as K becomes larger. This is quite different from the case for = 1, in which the ICS for K = 1 is much smaller than that for K = 0 and the 11

12 energy thresholds are close to each other. In Fig. 4(b), we present the ICSs for N in the J = Interestingly, as K increases, the ICS first increases, and then decreases. The energy threshold for K = 1 is visibly lower than that for K = 0 while the energy threshold for K = 2 is close to that for K = 0. The irregular effect of K on the ICS makes it difficult to interpret the rotational excitation based on simple physical models. Figure 4(c) shows the K-averaged ICSs for N in the J = 0 00, 1 11, and 2 11 states and in the ground rotational state. From this figure, it can be seen that the ICS for J = 1 11 is almost identical to that for J = 0 00 over the whole energy range studied. The ICS for J = 2 11 is, however, slightly smaller than that for J = In addition, the three initial states of N have very close reaction energy thresholds. Therefore, we conclude that rotational excitations of N have a negligible effect on the reaction, at least for low-lying rotational states. The quantum dynamical calculations with rotational excited reactants are still computationally expensive even under the spectator-bond and CS approximations. Thus, it is interesting to examine the validity of the JS approximation in predicting the rotational mode specificity in polyatomic reactions. Figure 5 shows the calculated JS cross sections from several rotationally excited states studied here. It can be seen the JS model correctly predicts that the rotational excitation of inhibits the reaction while the rotational excitations of N have little effect on the reaction. We also compared the JS cross sections with the CS counterparts for different rotationally 12

13 excited states. The level of agreement is similar to that in the ground state shown in Fig. 2. C. Sudden Vector Proection model The Sudden Vector Proection (SVP) model has been proposed by us recently to understand and predict mode specificity in direct reactions It is argued that for these reactions the intramolecular vibrational energy redistribution (IVR) of the reactants is often much slower than the collision time in these reactions. As a result, they can be treated in the sudden limit. Furthermore, the SVP model attributes the efficacy of a reactant mode in promoting the reaction to its coupling with the reaction coordinate at the transition state, which is quantified by the overlap between the reactant normal mode ( Q i ) and reaction coordinate vectors ( Q RC ): P Q Q [0,1]. It has been shown that for many gas phase and gas-surface i i RC reactions, the SVP model appears to be quite good for predicting the effects of reactant vibrational modes. 49 Indeed, the vibrational mode specificity for the title reaction has been confirmed by our recent calculations. 36 However, for rotational modes, it becomes much more difficult to predict their effects due apparently to angular momentum coupling Table II shows the calculated SVP values for the reactant rotational modes in this reaction. Interestingly, the rotational mode of is predicted to have a large proection on the reaction coordinate while those of N have essentially no coupling. 13

14 While the latter is consistent with the dynamical results presented above, the former is obviously different as the rotational excitation of is found to inhibit the reaction. To understand this difference, it is helpful to take a closer look at the SVP model. As mentioned earlier, both the reaction coordinate vector Q RC and the reactant vectors Q i are obtained without considering the total angular momentum. Indeed, the SVP prediction concerning the rotational effect of is consistent with the J tot =0 results presented in Fig. 6. The lack of enhancement in the ICS is clearly due to the angular momentum coupling for J tot >0. D. A simple model for predicting rotational mode specificity Exact quantum scattering calculations on the rotational mode specificity are challenging due to the formidable computational costs associated with the calculations for many partial waves, especially beyond tetra-atomic reactive systems. It is thus highly desirable to capture the characteristics of the reaction dynamics by a minimal amount of calculations. In quantum dynamical studies, we often attempt to understand the reaction dynamics by calculating the probabilities with vanishing total angular momentum J tot = 0. The physical idea here is that the reaction proceeds mainly following the minimum energy path, and the contributions from higher J tot can be approximately obtained. 47 This assumption is generally valid for vibrationally excited reactants. For rotationally excited reactants, this model loses some of its effectiveness. 6, 10, 12, This issue has seldom been examined in larger polyatomic 14

15 reactions. Here, we investigate the rotational mode specificity in terms of reaction probabilities and propose a simple model. Figure 7(a) shows the probabilities for the ( = 0, 1, 2) + N ( J = 0) reaction with J tot = 0 and K = 0 as a function of total energy. It is clear from the figure that as increases, the reaction probability firstly increases and then decreases. This is completely different from the behavior of the ICS, in which a notable inhibition effect is found. We note however that for nonzero values, K in the parity-adapted basis takes values from 0 to the minimum value of J and J tot. In the case J =0 00, we have J=. It is thus physically more reasonable to average the reaction probabilities over all possible K values. For example, if we denote the initial state by (J tot,, K), for =1, the K-averaged probability can be calculated from the K-specific probabilities P by the following formula ( Jtot,, K ) P ( P 2 P ) /3 K averaged (0,1,0) (1,1,1). The weight of two in the formula stems from the different parities of the system for nonzero K values. It should be pointed out here that the adacent J tot values give very close probabilities and thus it is physically acceptable to use P (0,1,0) in place of P (1,1,0) in the formula. Similarly, for = 2, we P ( P 2 P 2 P ) /5. The calculated K-averaged have K averaged (0,2,0) (1,2,1) (2,2,2) probabilities for = 0, 1, and 2 are presented in Fig. 7(b). Clearly, there exists an inhibition effect, thus consistent with the results presented above. As increases, the reaction probability decreases. The energy threshold is, however, nearly unchanged. This is in good accord with the behavior shown by the ICS. Thus, the 15

16 K-averaged probabilities give a correct prediction of the rotational mode specificity of. The reaction probabilities for the ( = 0) + N ( J = 0 00, 1 11, 2 11 ) reaction are plotted in Fig. 8(a) with J tot = 0 and K = 0 as a function of total energy. It can be seen that there does not exist a regular trend for the K-specific probabilities. The probability for J = 1 11 is much smaller than that for J = 0 00 in the entire energy range. The probability for J = 2 11 presents the highest energy threshold. However, it increases much faster than the other two initial states. Again, this differs distinctly from the behavior found in the ICS. Following the model proposed above, we calculate the K-averaged probabilities for J = 0 00, 1 11, and 2 11, as shown in Fig. 8(b). From this figure, we can now see that the probability for J = 1 11 is almost identical to that for J = 0 00 while the probability for J = 2 11 is slightly smaller. The three initial rotational states have nearly the same energy threshold. Thus, the rotational effects of N observed in the ICSs are well reproduced by the K-averaged probabilities. These results for the title reaction suggest that the dynamical features on the rotational effect can be predicted by the K-averaged reaction probabilities. If one is ust interested in understanding the rotational mode specificity, this simple method will greatly reduce the computational costs. The success of this inexpensive model is 16

17 based on sound physical principles. It would be interesting to test this method in other polyatomic reactions. IV. CONCLUSIONS The initial state selected time-dependent wave packet method was employed to study the effect of reactant rotational excitations on the reactivity within a reduced seven-dimensional model. The quantum dynamical calculations were carried out on a newly developed accurate global PES that was fitted to more than ~100,000 high-level ab initio points. It is shown that the rotational excitation in inhibits the reaction while the effects of N rotational excitations are minimal. The spectator-bond approximation was tested by comparing the reduced seven-dimensional cross section with the earlier full-dimensional result. It was found that the spectator-bond approximation, i.e. fixing the distances of the two NH bonds in the reactant N, works very well in the energy range studied. Furthermore, the CS and JS approximations were also examined within the seven-dimensional model. The CS and JS cross sections both follow the CC results well in the low collision energy range while they become visibly larger in the high energy range with the former much better. In addition, we have compared our quantum dynamical results with the predictions of the SVP model. Finally, we propose a simple model for predicting rotational mode specificity based on K-averaged probabilities. 17

18 ACKNOWLEDGEMENTS: We acknowledge financial support for US Department of Energy (Grant No. DE-FG02-05ER15694). The calculations are performed at the National Energy Research Scientific Computing (NERSC) Center. 18

19 Table I. Numerical parameters used in the wave packet calculations. (Atomic units are used unless stated otherwise.) Grid/basis range and size: R [1.5, 11.5], N =21, int r1 N =3 asymp r1 total N R =70, int N R =27 1max =26, 23max =22, 3max =20, l 2max =20, J max =32 Initial wave packet: Absorbing potential: 1 ( ) ( ) 2 2 1/ 4 ( RR0) / 2 ik0r R e e 2 R 0 =8.5, k 0 =0.65, δ=0.25 xxa n t ( ) xmax xa F e, x x x abs R a =9.0, α R =0.05, n R =2.5 r 1a =3.5, r 1 =0.035, a n r 1 =2.0 max F Flux position: r 1 =3.5 19

20 Table II. SVP values ( P i Q i Q RC ) for the +N reaction. Species SVP Mode 0.17 Rotation N Rotation 1 a Rotation 2 b a : Rotation around the C 2v axis. b : Rotation around the axis perpendicular to the C 2v axis. 20

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25 FIGURE LEGENDS Fig. 1: The nine-dimensional Jacobi coordinates for the AB + CDE system. Fig. 2: Comparison of 9D CS, 7D CS, CC and JS ICSs for the ( N ( J = 0) + = 0 00 ) reaction in the upper panel (a) and 7D CS and JS ICSs for the ( = 0, 2) + N ( J = 0 00, 2 11 ) reaction in the lower panel (b). Fig. 3: The K-specific and K-averaged CS ICSs for the ( ) + N ( J = 0 00 ) reaction. (a) = 1; (b) = 2; (c) = 0, 1, 2 and K-averaged. Fig. 4: The K-specific and K-averaged CS ICSs for the ( = 0) + N ( J ) reaction. (a) J = 1 11 ; (b) J = 2 11 ; (c) J = 0 00, 1 11, 2 11 and K-averaged. Fig. 5: The JS ICSs for the ( =0-2) + N ( J =0 00, 1 11, 2 11 ) reaction. Fig. 6: The reaction probabilities for the ( =0-2) + N ( J =0 00, 1 11, 2 11 ) reaction with total angular momentum J tot =0. Fig. 7: The probabilities for the ( = 0, 1, 2) + N ( J = 0 00 ) reaction as a function of total energy. (a) K = 0 and J tot = 0; (b) K-averaged. Fig. 8: The probabilities for the ( = 0) + N ( J = 0 00, 1 11, 2 11 ) reaction as a function of total energy. (a) K = 0 and J tot = 0; (b) K-averaged. 25

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