Article pubs.acs.org/JPCA
Vibrational and Rotational Mode Specificity in The Cl + H2O → HCl + OH Reaction: A Quantum Dynamical Study Hongwei Song and Hua Guo* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States ABSTRACT: The vibrational and rotational mode specificity of the Cl + H2O → HCl + OH reaction is studied on a recently constructed ab initio based global potential energy surface using an initial state selected Chebyshev real wave packet method. The full-dimensional quantum dynamical results under the centrifugal sudden and/or J-shifting approximations indicate that this reaction is enhanced strongly by excitations of the stretching modes of the H2O reactant but only weakly by bending excitations. On the other hand, combination modes are found to enhance the reaction more than the sum of individual excitations. In addition, rotational excitation of the H2O reactant slightly inhibits the reactivity. The observed mode specificity is consistent with the predictions of the recently proposed Sudden Vector Projection model, which attributes the promotional effects of the reactant modes to their couplings with the reaction coordinate at the transition state.
I. INTRODUCTION Recently, there has been much interest in understanding the dependence of reactivity of chemical reactions on reactant vibrational and/or rotational excitations.1−7 This so-called mode specificity,8 and related bond selectivity,9 not only shed valuable light on the dynamics of chemical reactions but also have practical implications in controlling the reactivity and product branching ratio. Based on the product energy disposal in atom−diatom reactions, Polanyi invoked microscopic reversibility and proposed two intuitive and useful rules concerning the relative efficacy of the translational and vibrational excitations in promoting such reactions based on the location of the barrier.10 For an early barrier reaction, for example, translational excitation is more effective in promoting the reaction than vibrational excitation. For a late barrier reaction, in contrast, vibrational excitation enhances the reaction more effectively than translational excitation. These empirical rules identify the location of the barrier as an important descriptor for mode specificity in atom−diatom reactions and offer a valuable guidance for polyatomic reactions. With the advent of laser techniques,11 it is now possible to prepare single quantum states in reactant molecules to explore mode specificity beyond atom−diatom reactions.6,8,12 For example, the mode specificity of the X + H2O (X = H, F, O(3P) and Cl) → HX + OH reactions has been experimentally probed by several authors.13−20 Understanding these experimental observations requires first-principles theory, namely accurate quantum dynamics on ab initio based global potential energy surfaces (PESs). Because only six internal degrees of freedom are involved, full-dimensional quantum scattering calculations are possible, albeit challenging. Unlike the more demanding atom-penta-atomic reactions,21−23 full-dimensional © 2015 American Chemical Society
quantum dynamical calculations for various X + H2O reactions24−38 have been reported by several groups including us using accurate global PESs.39−45 The six internal degrees of freedom endow the X + H2O reactions with much richer chemistry than atom−diatom reactions.7 For example, our full-dimensional quantum mechanical calculations on an accurate potential energy surface (PES) have shown that all three vibrational modes of the H2O reactant promote the F + H2O reaction more effectively than the translational mode.32 This observation is in contrast to the prediction of a naively extension of Polanyi’s rules, which anticipate the translational mode to be most effective because of the early barrier of this reaction. More recently, the reverse of the title reaction was also found to violet the extended Polanyi’s rules as the vibrational excitation of the HCl reactant enhances the reactivity more than the translational excitation, despite its early barrier.36 To rationalize these interesting and surprising findings, we proposed the Sudden Vector Projection (SVP) model.46 Instead of using the location of the barrier as a descriptor of mode specificity, the SVP model attributes the ability of a reactant mode in promoting the reaction to its coupling with the reaction coordinate at the transition state. This model has been shown to be consistent with Polanyi’s rules in atom−diatom reactions46 and correctly predicted the mode specificity observed for the F + H2O and HCl + OH reactions where the extended Polanyi’s rules failed.33,36 Since then, the SVP model has been applied to many reactions in the gas phase and at gas-surface interfaces with general Received: April 19, 2015 Revised: May 19, 2015 Published: May 19, 2015 6188
DOI: 10.1021/acs.jpca.5b03740 J. Phys. Chem. A 2015, 119, 6188−6194
Article
The Journal of Physical Chemistry A success.47−50 For the title reaction, the SVP model predicted that the stretching vibrational modes of H2O strongly promote the reaction, while the bending mode has roughly the same efficacy as the translational mode in enhancing the reactivity.33 In this publication, we perform the first full-dimensional quantum dynamical calculations on the title reaction on a recently developed PES and compare the results with the predictions of the SVP model. The results reported here, and those reported earlier on the bond selective Cl + HOD reaction,34 shed important light on the dynamics of this prototypical reaction. This publication is organized as follows. The next section (section II) outlines the scattering theory and numerical implementation. This is followed by section III with the results and discussion. The last section (section IV) concludes.
coupled BF total angular momentum eigenfunctions, which are defined as ε −1/2 Φ JM jK = (1 + δK 0)
j −K
+ ε( −1)l1+ j2 + j12 + J D−J *K , M Y l112j 2
+
j K
Y l112j = 2
+ V̂ (R , r1, r2 , θ1, θ2 , ϕ1) −
−
j * ∑ DKm (0, θ1, φ) 12
m
and yjm denotes the spherical harmonics. Note the restriction that ε(−1)l1+j2+j12+J = 1 for K = 0 in eq 5. For activated reactions, the centrifugal-sudden (CS) approximation52,53 is often quite appropriate. In the CS approximation, the Coriolis coupling is ignored and the centrifugal poten̂ )2 in the Hamiltonian, is given by tial, i.e. (J ̂ − j12 ε ̂ ̂ 2 JMε ⟨Φ JM jK |(J − j12 ) |Φ j ′ K ′⟩
V2ref (r2)
= δjj ′δKK ′[J(J + 1) + j12 (j12 + 1) − 2K 2]
2
where R is the distance between the attacking atom Cl and the H2O center of mass, r1 the distance between H and the center of mass of OH, and r2 the bond length of OH, with μR, μ1, and μ2 as their corresponding reduced masses. l1̂ is the orbital angular momentum operator of H with respect to OH, and j2̂ is the rotational angular momentum operator of OH, which ̂ . The one-dimensional (1D) reference are coupled to j12 Hamiltonians are defined as
(7)
k≥1
(8)
where |ψ1⟩ = DĤ scaled|ψ0⟩ and |ψ0⟩ = |χi⟩. To impose outgoing boundary conditions, the following Gaussian shaped damping function D is applied at the grid edges:
i = 1, 2 (2)
D(x) = e−α[(x − xa)/(xmax − xa)]
where V (ri) are the corresponding 1D reference potentials along ri. The parity (ε) adapted wave function is expanded as25
(9)
where x = R and r1, and xa is the starting point of the damping function. The scaled Hamiltonian is defined as Ĥ scaled = (Ĥ − H̅ )/ΔH to avoid the divergence of the Chebyshev propagator outside the range [−1,1]. The mean and half-width of the Hamiltonian were calculated from the spectral extrema Hmin and Hmax as H̅ = (Hmax + Hmin)/2 and ΔH = (Hmax − Hmin)/2. Since the initial wave packet is real, the Chebyshev iteration can be efficiently and accurately realized in real arithmetic.56,57 The action of the Hamiltonian is evaluated by transforming the wave function in the finite basis representation (FBR) and the discrete variable representation (DVR).58 The wave packet
ε v1 JMε ̂ FnJM ν1v2jK un (R )φν (r1)φν (r2)Φ jK (R , r1̂ , r2̂ ) 1
cos(kiR )|υ0j0 τ ; Jε⟩
|ψk + 1⟩ = D(2Ĥ scaled|ψk⟩ − D|ψk − 1⟩),
ref
∑
/2δ 2
where N is the normalization factor, R0 and δ are the mean position and width of the initial Gaussian function, and ki is the mean momentum given by Ei via ki = (2μREi)1/2. The initial rovibrational wave function of H2O was obtained by diagonalizing the three-dimensional Hamiltonian of the reactant, in which υ0, j0, and τ denote the initial vibrational quantum number, the initial angular momentum quantum number, and the parity of the reactant H2O, respectively. The wave packet is propagated using the Chebyshev propagator:54−56
Figure 1. Jacobi coordinates for the A + BCD system.
ψ JMε(R⃗ , r1⃗ , r2⃗ ) =
(6)
Since the coupling between different K blocks is neglected, K becomes a good quantum number and is conserved. The initial wave packet |χi⟩ is constructed as the direct product of a Gaussian wave packet in the scattering coordinate and a specific ro-vibrational state of H2O in the BF representation: |χi ⟩ = Ne−(R − R 0)
1 ∂2 + V ref (ri), 2μi ∂ri 2
(5)
2
(1)
hî (ri) = −
2l1 + 1 ⟨j2 ml10|j12 m⟩ 4π
yj m (θ2 , 0)
2
2μ2 r2 2
(4) 51
where are the Wigner rotation matrices. M is the projection of J on the z axis in the space-fixed (SF) frame, and j K K is the projection on the BF z axis that coincides with R. Yl112j2 are defined as
2 (J ̂ − j12̂ )2 l1̂ 1 ∂2 ̂ (r ) + h ̂ (r ) + + h Ĥ = − + 1 1 2 2 2μR ∂R2 2μR R2 2μ1r12
V1ref (r1)
]
DJK,M
II. THEORY The full-dimensional Hamiltonian in the atom-triatom Jacobi coordinates, as shown in Figure 1, for a given total angular momentum J is written as follows (ℏ = 1):
j2̂
2J + 1 J * j12 K [DK , M Y l1j 2 8π
2
n , ν1, v2j , K
(3)
where n labels the translational basis functions, ν1 and ν2 represent the vibrational basis indices for r1 and r2, and the composite index j denotes (l1, j2, j12). The translational ν basis functions, un1, are dependent on ν1 due to the use of an ̂ L-shaped grid. ΦJMε jK (R, r̂1, r̂2) in eq 3 are the parity-adapted 6189
DOI: 10.1021/acs.jpca.5b03740 J. Phys. Chem. A 2015, 119, 6188−6194
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The Journal of Physical Chemistry A is first prepared in FBR, in which the kinetic energy operator is diagonal in the CS approximation, and then transformed to DVR via one-dimensional pseudospectral transformation,59 where the potential energy operator is diagonal. The flux through the dividing surface, S = [r1 = rF1 ], was calculated from the energy-dependent scattering wave function, which is obtained by Fourier transforming the wave packet at the dividing surface.60 The initial state-selected total reaction probability is computed as follows: PυJ0εj τK 0(E) = 0
Table I. Numerical Parameters Used in the Wave Packet Calculationsa grid/basis range and size
R ∈ [2.0, 19.0] int Ntot R = 324, NR = 104 int asy Nr1 = 41, Nr1 = 9 b c asy int asy Nint r2 = 41, Nr2 = 9 /Nr2 = Nr2 = 6
1 2πμ1|ai(E)| (H −)2 sin 2 θ 2
initial wave packet damping term
j1max = 48, j2max = 36 R0 = 14.5, δ = 0.15, Ei = 0.65 eV Ra = 15.5, αR = 0.075, nR = 1.5, r1a = 4.0 αr1 = 0.05, nr1 = 1.5
flux position
rF1 = 4.0
a
× Im
∑ (2 − δk0)e
−ikθ
Atomic units are used unless stated otherwise. bTreating the two OH bonds in H2O equally. cTreating one OH bond as non-reactive in H2O.
ψk
k
⎡
⎤ ψk ′⎥ ∂r1 ⎦
∑ (2 − δk′ 0)e−ik′ θ⎢δ(r1 − r1F) ∂ ⎣
k′
(10)
where the Chebyshev angle θ = arccos Escaled is a nonlinear mapping of the scaled energy. The energy amplitude of the initial wave packet at the collision energy E is given by
ai(E) = ⟨ϕiE|χi ⟩
(11)
where |ϕiE⟩ is the free scattering wave function. The total reaction integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all relevant partial waves: συ0j τ(E) = 0
1 (2j0 + 1)
∑ K 0ε
π k2
∑
(2J + 1)PυJ0εj τK 0(E)
Figure 2. Calculated reaction probabilities for the Cl + H2O(0, 0, 0) → HCl + OH reaction as a function of translational energy with the total angular momentum J = 0. The models with different treatments of the spectator OH mode are compared.
0
J≥K0
(12)
Even using the CS approximation, it is still very challenging to carry out quantum scattering calculations on the title reaction due to the heavy atoms (Cl and O) involved in the system. A further approximation is to use the J-shifting (JS) approximation,61 in which the reaction probabilities from nonzero total angular momentum partial waves are all calculated by simply shifting the reaction probability from zero total angular momentum partial wave, i.e., PJ>0(E) = PJ=0(E − ΔE), where ΔE = B*J(J + 1). The rotational constant B* is approximately calculated by B* = 1/2μRR*2 with R* being the length of the scattering coordinate at the transition state. This approximation can significantly save the computational costs. However, it may result in considerable errors.
the same basis/grid number is employed along the coordinates r1 and r2. As a result, both OH bonds are allowed to break. Our results shown in Figure 2 indicate that the reaction probabilities from the two treatments are quite close except at higher energies. Thus, in all subsequent calculations, the treatment with the smaller basis is used. In addition, it should be noted that the reaction probability is extremely small from the reactant ground state over the energy range studied. The calculated CS integral cross sections (ICSs) from the lowest five vibrational states of H2O are plotted as a function of translational energy in Figure 3a and total energy in Figure 3b. Insets in the figure provide more details on the reaction thresholds. We can see from Figure 3a that excitations of the bending mode slightly promote the reactivity. The energy thresholds are shifted to lower energies and the probabilities become larger as n2 increases. On the other hand, excitation in either the symmetric or antisymmetric stretching mode significantly enhances the reaction. Although they have nearly the same energy threshold, the enhancement from excitation of the symmetric stretching mode is visibly larger than that of the antisymmetric stretching mode, and this difference increases with the collision energy. In Figure 3b where the ICSs are plotted in total energy, it can be seen that excitation in each of three vibrational modes is more effective than translational energy in promoting the reaction. This is because the ICSs from vibrationally excited states have not only lower energy thresholds but also larger magnitudes than that of the ground state over the entire energy range studied.
III. RESULTS AND DISCUSSION The numerical parameters employed in the calculations are given in Table I. The parameters were extensively tested to give converged results. The propagation requires around 3500 Chebyshev steps. A total of 260 partial waves were employed to converge the ICSs. We first examine the Cl + H2O(0, 0, 0) reaction with the total angular momentum J = 0. The vibrational state of H2O is denoted here by (n1, n2, n3), in which the three quantum numbers are for the symmetric stretching, bending, and antisymmetric stretching modes. The two curves in Figure 2 were obtained from two different treatments of the spectator OH bond in H2O. One used six vibrational basis functions for the spectator OH bond, thus making it impossible to dissociate. The other treated both OH bonds on equal footing, in which 6190
DOI: 10.1021/acs.jpca.5b03740 J. Phys. Chem. A 2015, 119, 6188−6194
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The Journal of Physical Chemistry A
Figure 3. Calculated CS ICSs for the Cl + H2O(n1, n2, n3) → HCl + OH reaction for the first five vibrationally excited states as a function of (a) translational energy and (b) total energy. Insets provide a close-up view of the reaction thresholds.
Figure 5. Calculated JS ICSs for the Cl + H2O → HCl + OH reaction for the first 11 vibrationally excited states as a function of (a) translational energy and (b) total energy.
Figure 5a, it can be seen that the ICS from the (1, 1, 0) state is not simply the sum of those from the (1, 0, 0) and (0, 1, 0) states at a specific collision energy. The combined excitation of the symmetric stretching and bending modes enhances the reaction much more than the sum of individual excitations. Further excitation to the (1, 2, 0) level of H2O results in additional reactivity. Similarly, the same behavior is observed for the combined excitation of the antisymmetric stretching and bending modes. These interesting observations suggest some synergistic effect between the stretching and bending modes. In other words, bending excitations seem to result in significant enhancement only when accompanied by stretching excitations. From Figure 5b, in which the ICSs are plotted as a function of total energy, it can be seen that the excitation of combination modes has a negligible effect on their energy threshold, but they are much more effective than translational energy in promoting the reactivity. The mode specificity observed in this late-barrier reaction resembles the much more extensively studied H + H2O → H2 + OH reaction. For the H + H2O → H2 + OH reaction, it was reported by Fu and Zhang30 that excitation of the bending mode is less effective in promoting the reactivity than the translational mode, while the excitation of individual symmetric or antisymmetric stretching mode greatly enhances the reaction. In addition, the enhancement effect of the symmetric stretching mode excitation is slightly larger than the antisymmetric stretching mode excitation. These dynamical features on the mode specificity are quite similar to what we observed here for the Cl + H2O → HCl + OH reaction. However, they found that combination excitation of the stretching and bending modes showed comparable enhancement with individual stretching mode excitations, i.e. it is almost equal to the summing effect of individual excitation of the two. The different behaviors are
Figure 4. Comparison of the calculated CS and JS ICSs for the Cl + H2O → HCl + OH reaction for the first five vibrationally excited states as a function of translational energy.
Figure 4 shows the calculated CS and JS ICSs for the title reaction from the first five vibrationally excited states. Clearly, the JS ICSs reproduce the CS counterparts quite well. The differences between the two sets of results are almost negligible at low collision energies while become slightly larger at high collision energies. In addition, the discrepancy between the two approximations does not increase with the initial excitation of the H2O reactant. Therefore, we used the JS model to compute the ICSs for higher vibrational states of H2O. The calculated JS ICSs from the lowest 11 vibrational excited states are displayed in Figure 5 as a function of (a) translational energy and (b) total energy, respectively. It is interesting to examine the effect of excitations of the H2O combination bands. By comparing the ICSs from the (1, 1, 0) state and from the (1, 0, 0) and (0, 1, 0) states in 6191
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Table II. SVP Values (ηi = Q⃗ i·Q⃗ RC) for the Cl + H2O Reaction
probably due to the intricate details of the PESs and different mass combinations for these two reactions. The rotational mode specificity of the title reaction was studied by exciting the H2O reactant to different rotational states. Figure 6 shows the calculated JS ICSs for the Cl + H2O
species
SVP
υas υss υb rot. I rot. II trans.
0.70 0.71 0.05 0.02 0.03 0.10
Table II shows the calculated SVP values for the title reaction. It can be seen that the symmetric stretching mode of H2O has the largest projection on the reaction coordinate at the transition state, which is followed closely by the antisymmetric stretching mode. So, they are both strongly coupled with the reaction coordinate, implying large enhancement effects. As the SVP values for the two stretching modes are larger than that of the translational mode, their initial excitations are more effective than translation energy in promoting the reaction. This is consistent with the quantum dynamical results. The bending mode has a SVP value of 0.05, which is smaller than the translational one of 0.1. This prediction is consistent with the small promotional effect of the bending mode, but underestimates its relative enhancement to the translational excitation. This under prediction is presumably caused by the low frequency of the bending mode, which makes the IVR more efficient.49 Unfortunately, the SVP model does not provide any guidance on the effects of combination modes. For the two H2O rotational degrees of freedom, the SVP values are 0.02 and 0.03, which suggest that they are weakly coupled with the reaction coordinate. This is consistent with the quantum results, in which rotational excitations of H2O do not promote the reaction. Note that the SVP model cannot predict inhibitory effects.
Figure 6. Calculated JS ICSs for the Cl + H2O → HCl + OH reaction for first several rotationally excited states as a function of (a) translational energy and (b) total energy.
IV. CONCLUSIONS The mode specificity of the Cl + H2O → HCl + OH reaction is investigated using the initial state-selected Chebyshev real wave packet method on an accurate ab initio based global PES. These quantum dynamical calculations indicate that excitations in either the symmetric or antisymmetric stretching modes of the H2O reactant greatly promote the reaction with the former having a higher efficacy. On the other hand, excitations of the H2O bending mode alone enhance the reaction only slightly. Interestingly, bending excitations accompanied by stretching excitations provide the strongest enhancement. All vibrational excitations promote the reactivity more effectively than translational energy. In addition, rotational excitations of H2O inhibit the reaction. Finally, these quantum dynamical results are rationalized by the recently proposed SVP model.
(JKaKc = 000, 101, 111, 110, 303, 322, 330) → HCl + OH reaction as a function of (a) translational energy and (b) total energy. Apparently, rotational excitations of H2O have a negligible effect on the reaction energy threshold while a notable effect on the reactivity. The ICS generally decreases when the H2O reactant is excited to a higher rotational state. Thus, excitation of the rotational mode of H2O inhibits the reaction and the inhibitory effect increases with the collision energy. On the other hand, it appears that the 101 (301) excitation function is close to the 111 (322) excitation function, which are both slightly larger than the 110 (310) excitation function. From Figure 6b, in which the ICS is plotted as a function of total energy, it can be seen that rotational excitations of H2O are less effective than translation energy in promoting the reaction. To understand the mode specificity observed above for the title reaction, we relied on the recently proposed SVP model.49 The SVP model assumes that the intramolecular vibrational energy redistribution (IVR) of reactants is much slower than the collision time, i.e. the reaction takes place in the sudden limit.46 This is a reasonably assumption for this reaction as the density of states is quite sparse for H2O, leading to slow IVR.62 In the SVP model, the efficacy of a reactant mode in promoting the reaction is attributed to its coupling with the reaction coordinate at the transition state, which is quantified by the projection of the reactant normal mode vector (Q⃗ i) onto the reaction coordinate vector (Q⃗ RC): ηi = Q⃗ i · Q⃗ RC ∈ [0,1].
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by Department of Energy (DE-FG0205ER15694 to HG) and calculations were performed at the National Energy Research Scientific Computing (NERSC) Center. 6192
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Vibrational and Rotational Mode Specificity in The Cl + H2O → HCl + OH Reaction: A Quantum Dynamical Study Hongwei Song and Hua Guo* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States ABSTRACT: The vibrational and rotational mode specificity of the Cl + H2O → HCl + OH reaction is studied on a recently constructed ab initio based global potential energy surface using an initial state selected Chebyshev real wave packet method. The full-dimensional quantum dynamical results under the centrifugal sudden and/or J-shifting approximations indicate that this reaction is enhanced strongly by excitations of the stretching modes of the H2O reactant but only weakly by bending excitations. On the other hand, combination modes are found to enhance the reaction more than the sum of individual excitations. In addition, rotational excitation of the H2O reactant slightly inhibits the reactivity. The observed mode specificity is consistent with the predictions of the recently proposed Sudden Vector Projection model, which attributes the promotional effects of the reactant modes to their couplings with the reaction coordinate at the transition state.
I. INTRODUCTION Recently, there has been much interest in understanding the dependence of reactivity of chemical reactions on reactant vibrational and/or rotational excitations.1−7 This so-called mode specificity,8 and related bond selectivity,9 not only shed valuable light on the dynamics of chemical reactions but also have practical implications in controlling the reactivity and product branching ratio. Based on the product energy disposal in atom−diatom reactions, Polanyi invoked microscopic reversibility and proposed two intuitive and useful rules concerning the relative efficacy of the translational and vibrational excitations in promoting such reactions based on the location of the barrier.10 For an early barrier reaction, for example, translational excitation is more effective in promoting the reaction than vibrational excitation. For a late barrier reaction, in contrast, vibrational excitation enhances the reaction more effectively than translational excitation. These empirical rules identify the location of the barrier as an important descriptor for mode specificity in atom−diatom reactions and offer a valuable guidance for polyatomic reactions. With the advent of laser techniques,11 it is now possible to prepare single quantum states in reactant molecules to explore mode specificity beyond atom−diatom reactions.6,8,12 For example, the mode specificity of the X + H2O (X = H, F, O(3P) and Cl) → HX + OH reactions has been experimentally probed by several authors.13−20 Understanding these experimental observations requires first-principles theory, namely accurate quantum dynamics on ab initio based global potential energy surfaces (PESs). Because only six internal degrees of freedom are involved, full-dimensional quantum scattering calculations are possible, albeit challenging. Unlike the more demanding atom-penta-atomic reactions,21−23 full-dimensional © 2015 American Chemical Society
quantum dynamical calculations for various X + H2O reactions24−38 have been reported by several groups including us using accurate global PESs.39−45 The six internal degrees of freedom endow the X + H2O reactions with much richer chemistry than atom−diatom reactions.7 For example, our full-dimensional quantum mechanical calculations on an accurate potential energy surface (PES) have shown that all three vibrational modes of the H2O reactant promote the F + H2O reaction more effectively than the translational mode.32 This observation is in contrast to the prediction of a naively extension of Polanyi’s rules, which anticipate the translational mode to be most effective because of the early barrier of this reaction. More recently, the reverse of the title reaction was also found to violet the extended Polanyi’s rules as the vibrational excitation of the HCl reactant enhances the reactivity more than the translational excitation, despite its early barrier.36 To rationalize these interesting and surprising findings, we proposed the Sudden Vector Projection (SVP) model.46 Instead of using the location of the barrier as a descriptor of mode specificity, the SVP model attributes the ability of a reactant mode in promoting the reaction to its coupling with the reaction coordinate at the transition state. This model has been shown to be consistent with Polanyi’s rules in atom−diatom reactions46 and correctly predicted the mode specificity observed for the F + H2O and HCl + OH reactions where the extended Polanyi’s rules failed.33,36 Since then, the SVP model has been applied to many reactions in the gas phase and at gas-surface interfaces with general Received: April 19, 2015 Revised: May 19, 2015 Published: May 19, 2015 6188
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The Journal of Physical Chemistry A success.47−50 For the title reaction, the SVP model predicted that the stretching vibrational modes of H2O strongly promote the reaction, while the bending mode has roughly the same efficacy as the translational mode in enhancing the reactivity.33 In this publication, we perform the first full-dimensional quantum dynamical calculations on the title reaction on a recently developed PES and compare the results with the predictions of the SVP model. The results reported here, and those reported earlier on the bond selective Cl + HOD reaction,34 shed important light on the dynamics of this prototypical reaction. This publication is organized as follows. The next section (section II) outlines the scattering theory and numerical implementation. This is followed by section III with the results and discussion. The last section (section IV) concludes.
coupled BF total angular momentum eigenfunctions, which are defined as ε −1/2 Φ JM jK = (1 + δK 0)
j −K
+ ε( −1)l1+ j2 + j12 + J D−J *K , M Y l112j 2
+
j K
Y l112j = 2
+ V̂ (R , r1, r2 , θ1, θ2 , ϕ1) −
−
j * ∑ DKm (0, θ1, φ) 12
m
and yjm denotes the spherical harmonics. Note the restriction that ε(−1)l1+j2+j12+J = 1 for K = 0 in eq 5. For activated reactions, the centrifugal-sudden (CS) approximation52,53 is often quite appropriate. In the CS approximation, the Coriolis coupling is ignored and the centrifugal poten̂ )2 in the Hamiltonian, is given by tial, i.e. (J ̂ − j12 ε ̂ ̂ 2 JMε ⟨Φ JM jK |(J − j12 ) |Φ j ′ K ′⟩
V2ref (r2)
= δjj ′δKK ′[J(J + 1) + j12 (j12 + 1) − 2K 2]
2
where R is the distance between the attacking atom Cl and the H2O center of mass, r1 the distance between H and the center of mass of OH, and r2 the bond length of OH, with μR, μ1, and μ2 as their corresponding reduced masses. l1̂ is the orbital angular momentum operator of H with respect to OH, and j2̂ is the rotational angular momentum operator of OH, which ̂ . The one-dimensional (1D) reference are coupled to j12 Hamiltonians are defined as
(7)
k≥1
(8)
where |ψ1⟩ = DĤ scaled|ψ0⟩ and |ψ0⟩ = |χi⟩. To impose outgoing boundary conditions, the following Gaussian shaped damping function D is applied at the grid edges:
i = 1, 2 (2)
D(x) = e−α[(x − xa)/(xmax − xa)]
where V (ri) are the corresponding 1D reference potentials along ri. The parity (ε) adapted wave function is expanded as25
(9)
where x = R and r1, and xa is the starting point of the damping function. The scaled Hamiltonian is defined as Ĥ scaled = (Ĥ − H̅ )/ΔH to avoid the divergence of the Chebyshev propagator outside the range [−1,1]. The mean and half-width of the Hamiltonian were calculated from the spectral extrema Hmin and Hmax as H̅ = (Hmax + Hmin)/2 and ΔH = (Hmax − Hmin)/2. Since the initial wave packet is real, the Chebyshev iteration can be efficiently and accurately realized in real arithmetic.56,57 The action of the Hamiltonian is evaluated by transforming the wave function in the finite basis representation (FBR) and the discrete variable representation (DVR).58 The wave packet
ε v1 JMε ̂ FnJM ν1v2jK un (R )φν (r1)φν (r2)Φ jK (R , r1̂ , r2̂ ) 1
cos(kiR )|υ0j0 τ ; Jε⟩
|ψk + 1⟩ = D(2Ĥ scaled|ψk⟩ − D|ψk − 1⟩),
ref
∑
/2δ 2
where N is the normalization factor, R0 and δ are the mean position and width of the initial Gaussian function, and ki is the mean momentum given by Ei via ki = (2μREi)1/2. The initial rovibrational wave function of H2O was obtained by diagonalizing the three-dimensional Hamiltonian of the reactant, in which υ0, j0, and τ denote the initial vibrational quantum number, the initial angular momentum quantum number, and the parity of the reactant H2O, respectively. The wave packet is propagated using the Chebyshev propagator:54−56
Figure 1. Jacobi coordinates for the A + BCD system.
ψ JMε(R⃗ , r1⃗ , r2⃗ ) =
(6)
Since the coupling between different K blocks is neglected, K becomes a good quantum number and is conserved. The initial wave packet |χi⟩ is constructed as the direct product of a Gaussian wave packet in the scattering coordinate and a specific ro-vibrational state of H2O in the BF representation: |χi ⟩ = Ne−(R − R 0)
1 ∂2 + V ref (ri), 2μi ∂ri 2
(5)
2
(1)
hî (ri) = −
2l1 + 1 ⟨j2 ml10|j12 m⟩ 4π
yj m (θ2 , 0)
2
2μ2 r2 2
(4) 51
where are the Wigner rotation matrices. M is the projection of J on the z axis in the space-fixed (SF) frame, and j K K is the projection on the BF z axis that coincides with R. Yl112j2 are defined as
2 (J ̂ − j12̂ )2 l1̂ 1 ∂2 ̂ (r ) + h ̂ (r ) + + h Ĥ = − + 1 1 2 2 2μR ∂R2 2μR R2 2μ1r12
V1ref (r1)
]
DJK,M
II. THEORY The full-dimensional Hamiltonian in the atom-triatom Jacobi coordinates, as shown in Figure 1, for a given total angular momentum J is written as follows (ℏ = 1):
j2̂
2J + 1 J * j12 K [DK , M Y l1j 2 8π
2
n , ν1, v2j , K
(3)
where n labels the translational basis functions, ν1 and ν2 represent the vibrational basis indices for r1 and r2, and the composite index j denotes (l1, j2, j12). The translational ν basis functions, un1, are dependent on ν1 due to the use of an ̂ L-shaped grid. ΦJMε jK (R, r̂1, r̂2) in eq 3 are the parity-adapted 6189
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The Journal of Physical Chemistry A is first prepared in FBR, in which the kinetic energy operator is diagonal in the CS approximation, and then transformed to DVR via one-dimensional pseudospectral transformation,59 where the potential energy operator is diagonal. The flux through the dividing surface, S = [r1 = rF1 ], was calculated from the energy-dependent scattering wave function, which is obtained by Fourier transforming the wave packet at the dividing surface.60 The initial state-selected total reaction probability is computed as follows: PυJ0εj τK 0(E) = 0
Table I. Numerical Parameters Used in the Wave Packet Calculationsa grid/basis range and size
R ∈ [2.0, 19.0] int Ntot R = 324, NR = 104 int asy Nr1 = 41, Nr1 = 9 b c asy int asy Nint r2 = 41, Nr2 = 9 /Nr2 = Nr2 = 6
1 2πμ1|ai(E)| (H −)2 sin 2 θ 2
initial wave packet damping term
j1max = 48, j2max = 36 R0 = 14.5, δ = 0.15, Ei = 0.65 eV Ra = 15.5, αR = 0.075, nR = 1.5, r1a = 4.0 αr1 = 0.05, nr1 = 1.5
flux position
rF1 = 4.0
a
× Im
∑ (2 − δk0)e
−ikθ
Atomic units are used unless stated otherwise. bTreating the two OH bonds in H2O equally. cTreating one OH bond as non-reactive in H2O.
ψk
k
⎡
⎤ ψk ′⎥ ∂r1 ⎦
∑ (2 − δk′ 0)e−ik′ θ⎢δ(r1 − r1F) ∂ ⎣
k′
(10)
where the Chebyshev angle θ = arccos Escaled is a nonlinear mapping of the scaled energy. The energy amplitude of the initial wave packet at the collision energy E is given by
ai(E) = ⟨ϕiE|χi ⟩
(11)
where |ϕiE⟩ is the free scattering wave function. The total reaction integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all relevant partial waves: συ0j τ(E) = 0
1 (2j0 + 1)
∑ K 0ε
π k2
∑
(2J + 1)PυJ0εj τK 0(E)
Figure 2. Calculated reaction probabilities for the Cl + H2O(0, 0, 0) → HCl + OH reaction as a function of translational energy with the total angular momentum J = 0. The models with different treatments of the spectator OH mode are compared.
0
J≥K0
(12)
Even using the CS approximation, it is still very challenging to carry out quantum scattering calculations on the title reaction due to the heavy atoms (Cl and O) involved in the system. A further approximation is to use the J-shifting (JS) approximation,61 in which the reaction probabilities from nonzero total angular momentum partial waves are all calculated by simply shifting the reaction probability from zero total angular momentum partial wave, i.e., PJ>0(E) = PJ=0(E − ΔE), where ΔE = B*J(J + 1). The rotational constant B* is approximately calculated by B* = 1/2μRR*2 with R* being the length of the scattering coordinate at the transition state. This approximation can significantly save the computational costs. However, it may result in considerable errors.
the same basis/grid number is employed along the coordinates r1 and r2. As a result, both OH bonds are allowed to break. Our results shown in Figure 2 indicate that the reaction probabilities from the two treatments are quite close except at higher energies. Thus, in all subsequent calculations, the treatment with the smaller basis is used. In addition, it should be noted that the reaction probability is extremely small from the reactant ground state over the energy range studied. The calculated CS integral cross sections (ICSs) from the lowest five vibrational states of H2O are plotted as a function of translational energy in Figure 3a and total energy in Figure 3b. Insets in the figure provide more details on the reaction thresholds. We can see from Figure 3a that excitations of the bending mode slightly promote the reactivity. The energy thresholds are shifted to lower energies and the probabilities become larger as n2 increases. On the other hand, excitation in either the symmetric or antisymmetric stretching mode significantly enhances the reaction. Although they have nearly the same energy threshold, the enhancement from excitation of the symmetric stretching mode is visibly larger than that of the antisymmetric stretching mode, and this difference increases with the collision energy. In Figure 3b where the ICSs are plotted in total energy, it can be seen that excitation in each of three vibrational modes is more effective than translational energy in promoting the reaction. This is because the ICSs from vibrationally excited states have not only lower energy thresholds but also larger magnitudes than that of the ground state over the entire energy range studied.
III. RESULTS AND DISCUSSION The numerical parameters employed in the calculations are given in Table I. The parameters were extensively tested to give converged results. The propagation requires around 3500 Chebyshev steps. A total of 260 partial waves were employed to converge the ICSs. We first examine the Cl + H2O(0, 0, 0) reaction with the total angular momentum J = 0. The vibrational state of H2O is denoted here by (n1, n2, n3), in which the three quantum numbers are for the symmetric stretching, bending, and antisymmetric stretching modes. The two curves in Figure 2 were obtained from two different treatments of the spectator OH bond in H2O. One used six vibrational basis functions for the spectator OH bond, thus making it impossible to dissociate. The other treated both OH bonds on equal footing, in which 6190
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The Journal of Physical Chemistry A
Figure 3. Calculated CS ICSs for the Cl + H2O(n1, n2, n3) → HCl + OH reaction for the first five vibrationally excited states as a function of (a) translational energy and (b) total energy. Insets provide a close-up view of the reaction thresholds.
Figure 5. Calculated JS ICSs for the Cl + H2O → HCl + OH reaction for the first 11 vibrationally excited states as a function of (a) translational energy and (b) total energy.
Figure 5a, it can be seen that the ICS from the (1, 1, 0) state is not simply the sum of those from the (1, 0, 0) and (0, 1, 0) states at a specific collision energy. The combined excitation of the symmetric stretching and bending modes enhances the reaction much more than the sum of individual excitations. Further excitation to the (1, 2, 0) level of H2O results in additional reactivity. Similarly, the same behavior is observed for the combined excitation of the antisymmetric stretching and bending modes. These interesting observations suggest some synergistic effect between the stretching and bending modes. In other words, bending excitations seem to result in significant enhancement only when accompanied by stretching excitations. From Figure 5b, in which the ICSs are plotted as a function of total energy, it can be seen that the excitation of combination modes has a negligible effect on their energy threshold, but they are much more effective than translational energy in promoting the reactivity. The mode specificity observed in this late-barrier reaction resembles the much more extensively studied H + H2O → H2 + OH reaction. For the H + H2O → H2 + OH reaction, it was reported by Fu and Zhang30 that excitation of the bending mode is less effective in promoting the reactivity than the translational mode, while the excitation of individual symmetric or antisymmetric stretching mode greatly enhances the reaction. In addition, the enhancement effect of the symmetric stretching mode excitation is slightly larger than the antisymmetric stretching mode excitation. These dynamical features on the mode specificity are quite similar to what we observed here for the Cl + H2O → HCl + OH reaction. However, they found that combination excitation of the stretching and bending modes showed comparable enhancement with individual stretching mode excitations, i.e. it is almost equal to the summing effect of individual excitation of the two. The different behaviors are
Figure 4. Comparison of the calculated CS and JS ICSs for the Cl + H2O → HCl + OH reaction for the first five vibrationally excited states as a function of translational energy.
Figure 4 shows the calculated CS and JS ICSs for the title reaction from the first five vibrationally excited states. Clearly, the JS ICSs reproduce the CS counterparts quite well. The differences between the two sets of results are almost negligible at low collision energies while become slightly larger at high collision energies. In addition, the discrepancy between the two approximations does not increase with the initial excitation of the H2O reactant. Therefore, we used the JS model to compute the ICSs for higher vibrational states of H2O. The calculated JS ICSs from the lowest 11 vibrational excited states are displayed in Figure 5 as a function of (a) translational energy and (b) total energy, respectively. It is interesting to examine the effect of excitations of the H2O combination bands. By comparing the ICSs from the (1, 1, 0) state and from the (1, 0, 0) and (0, 1, 0) states in 6191
DOI: 10.1021/acs.jpca.5b03740 J. Phys. Chem. A 2015, 119, 6188−6194
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Table II. SVP Values (ηi = Q⃗ i·Q⃗ RC) for the Cl + H2O Reaction
probably due to the intricate details of the PESs and different mass combinations for these two reactions. The rotational mode specificity of the title reaction was studied by exciting the H2O reactant to different rotational states. Figure 6 shows the calculated JS ICSs for the Cl + H2O
species
SVP
υas υss υb rot. I rot. II trans.
0.70 0.71 0.05 0.02 0.03 0.10
Table II shows the calculated SVP values for the title reaction. It can be seen that the symmetric stretching mode of H2O has the largest projection on the reaction coordinate at the transition state, which is followed closely by the antisymmetric stretching mode. So, they are both strongly coupled with the reaction coordinate, implying large enhancement effects. As the SVP values for the two stretching modes are larger than that of the translational mode, their initial excitations are more effective than translation energy in promoting the reaction. This is consistent with the quantum dynamical results. The bending mode has a SVP value of 0.05, which is smaller than the translational one of 0.1. This prediction is consistent with the small promotional effect of the bending mode, but underestimates its relative enhancement to the translational excitation. This under prediction is presumably caused by the low frequency of the bending mode, which makes the IVR more efficient.49 Unfortunately, the SVP model does not provide any guidance on the effects of combination modes. For the two H2O rotational degrees of freedom, the SVP values are 0.02 and 0.03, which suggest that they are weakly coupled with the reaction coordinate. This is consistent with the quantum results, in which rotational excitations of H2O do not promote the reaction. Note that the SVP model cannot predict inhibitory effects.
Figure 6. Calculated JS ICSs for the Cl + H2O → HCl + OH reaction for first several rotationally excited states as a function of (a) translational energy and (b) total energy.
IV. CONCLUSIONS The mode specificity of the Cl + H2O → HCl + OH reaction is investigated using the initial state-selected Chebyshev real wave packet method on an accurate ab initio based global PES. These quantum dynamical calculations indicate that excitations in either the symmetric or antisymmetric stretching modes of the H2O reactant greatly promote the reaction with the former having a higher efficacy. On the other hand, excitations of the H2O bending mode alone enhance the reaction only slightly. Interestingly, bending excitations accompanied by stretching excitations provide the strongest enhancement. All vibrational excitations promote the reactivity more effectively than translational energy. In addition, rotational excitations of H2O inhibit the reaction. Finally, these quantum dynamical results are rationalized by the recently proposed SVP model.
(JKaKc = 000, 101, 111, 110, 303, 322, 330) → HCl + OH reaction as a function of (a) translational energy and (b) total energy. Apparently, rotational excitations of H2O have a negligible effect on the reaction energy threshold while a notable effect on the reactivity. The ICS generally decreases when the H2O reactant is excited to a higher rotational state. Thus, excitation of the rotational mode of H2O inhibits the reaction and the inhibitory effect increases with the collision energy. On the other hand, it appears that the 101 (301) excitation function is close to the 111 (322) excitation function, which are both slightly larger than the 110 (310) excitation function. From Figure 6b, in which the ICS is plotted as a function of total energy, it can be seen that rotational excitations of H2O are less effective than translation energy in promoting the reaction. To understand the mode specificity observed above for the title reaction, we relied on the recently proposed SVP model.49 The SVP model assumes that the intramolecular vibrational energy redistribution (IVR) of reactants is much slower than the collision time, i.e. the reaction takes place in the sudden limit.46 This is a reasonably assumption for this reaction as the density of states is quite sparse for H2O, leading to slow IVR.62 In the SVP model, the efficacy of a reactant mode in promoting the reaction is attributed to its coupling with the reaction coordinate at the transition state, which is quantified by the projection of the reactant normal mode vector (Q⃗ i) onto the reaction coordinate vector (Q⃗ RC): ηi = Q⃗ i · Q⃗ RC ∈ [0,1].
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by Department of Energy (DE-FG0205ER15694 to HG) and calculations were performed at the National Energy Research Scientific Computing (NERSC) Center. 6192
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