Mode specificity in the HF + OH → F + H2O reaction Hongwei Song, Jun Li, and Hua Guo Citation: The Journal of Chemical Physics 141, 164316 (2014); doi: 10.1063/1.4900445 View online: http://dx.doi.org/10.1063/1.4900445 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quasiclassical trajectory study of the effect of antisymmetric stretch mode excitation on the O(3P) + CH4(ν3 = 1) → OH + CH3 reaction on an analytical potential energy surface. Comparison with experiment J. Chem. Phys. 141, 094307 (2014); 10.1063/1.4893988 Effects of reactant rotation on the dynamics of the OH + CH4 → H2O + CH3 reaction: A six-dimensional study J. Chem. Phys. 140, 084307 (2014); 10.1063/1.4866426 Dynamics study of the OH + NH3 hydrogen abstraction reaction using QCT calculations based on an analytical potential energy surface J. Chem. Phys. 138, 214306 (2013); 10.1063/1.4808109 Communication: Quasiclassical trajectory calculations of correlated product-state distributions for the dissociation of (H2O)2 and (D2O)2 J. Chem. Phys. 135, 151102 (2011); 10.1063/1.3655564 Communication: State-to-state quantum dynamics study of the OH + CO → H + CO2 reaction in full dimensions (J = 0) J. Chem. Phys. 135, 141108 (2011); 10.1063/1.3653787
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THE JOURNAL OF CHEMICAL PHYSICS 141, 164316 (2014)
Mode specificity in the HF + OH → F + H2 O reaction Hongwei Song, Jun Li, and Hua Guoa) Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA
(Received 29 August 2014; accepted 15 October 2014; published online 28 October 2014) Full-dimensional quantum dynamics and quasi-classical trajectory calculations are reported for the title reaction on a recently constructed ab initio based global potential energy surface. Strong mode specificity was found, consistent with the prediction of the sudden vector projection model. Specifically, the HF vibration strongly promotes the reaction while the OH vibration has little effect. Rotational excitations of both reactants slightly enhance the reaction. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4900445] I. INTRODUCTION
Reactions involving four atoms present an ideal proving ground to study reaction dynamics.1–3 This is because such systems offer much richer complexity than atom-diatom reactions but are sufficiently small to be amenable to fulldimensional quantum dynamical characterization on accurate global potential energy surfaces (PESs).4–6 With six degrees of freedom, mode specificity in these systems provides valuable insights into reaction dynamics.7 In this work, we focus on the F + H2 O → HF + OH reaction. Experimentally, the products have been found to be dominated by strong HF vibrational excitation while OH has essentially no internal excitation, behaving as a spectator.8–12 Evidence is also presented for non-adiabatic transitions between the lowest two electronic states.12 Theoretically, ab initio studies revealed that the reaction path on the lowest electronic state features a low and “early” barrier, flanked by significant pre- and post-transition state wells.13, 14 Several full-dimensional global PESs have been developed based on large numbers of high-level ab initio points,15, 16 cumulating with an externally scaled PES that has the correct exothermicity and barrier height17 and is capable of reproducing the measured cross section.18 Quantum dynamics (QD) and quasi-classical trajectory (QCT) studies on these PESs have revealed many surprising and interesting behaviors. For instance, it was found that the hemi-bonded pre-transition state well19 significantly enhances the reactivity at low collision energies by steering the reactants towards the barrier.20 Furthermore, excitations of all vibrational modes of H2 O were found to promote the reaction more effectively than translational energy.21 The mode specificity was surprising as the reactant vibrational modes were expected to have lower efficacies than translation for this “early” barrier reaction, according to Polanyi’s Rules.22 Furthermore, rotational excitations of the H2 O reactant were also shown to enhance reactivity substantially.18 These vibrational and rotational mode specific effects can nonetheless be explained by the recently proposed Sudden Vector Projection (SVP) model,23 which ata) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/141(16)/164316/7/$30.00
tributes the mode specificity to the coupling between the reactant modes and the reaction coordinate at the transition state.24 In this work, we investigate the HF + OH → F + H2 O reaction, which is the reverse of the much studied F + H2 O reaction. While this endothermic reaction has been investigated neither experimentally nor theoretically, it presents a prototype to validate the SVP model for diatom-diatom reactions. In particular, we examine in this work effects of reactant vibrational and rotational excitations on the overall reactivity using both QD and QCT methods on an accurate PES. We note in particular that unlike the state-to-state S-matrix elements, the initial state specific cross sections reported here cannot be directly obtained from detailed balance, although microscopic reversibility does provide some qualitative insights for reactivities in both directions of the reaction. This article is organized as follows. Section II details the theoretical methods used to investigate the reaction dynamics and mode specificity. The results are presented and discussed in Sec. III. Finally, conclusions are given in Sec. IV. II. THEORY A. Quantum dynamics
The full-dimensional Hamiltonian for the diatom-diatom reaction in the reactant Jacobi coordinates, as shown in Fig. 1, for a given total angular momentum J can be written as (¯ = 1 thereafter)25 ˆ ˆ 2 1 ∂2 ˆ 1 (r1 ) + hˆ 2 (r2 ) + (J − j12 ) Hˆ = − + h 2μR ∂R 2 2μR R 2 +
jˆ12 jˆ22 + + Vˆ (R, r1 , r2 , θ1 , θ2 , ϕ1 ) 2μ1 r12 2μ2 r22 ref
ref
− V1 (r1 ) − V2 (r2 ),
(1)
where r1 , r2 , and R are the bond distances of AB and CD, and the distance between their centers of mass (COMs), respectively, with μ1 , μ2 , and μR as their corresponding reduced masses. jˆ1 and jˆ2 are the rotational angular momentum operators of AB and CD, which are coupled to jˆ12 . Jˆ denotes the total angular momentum of the system. The one-dimensional
141, 164316-1
© 2014 AIP Publishing LLC
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the BF representation, is given by J Mε j K (Jˆ − jˆ12 )2 Jj Mε K = δjj δKK [J (J + 1) +j12 (j12 + 1) − 2K 2 ] + 1/2 −δK+1K λ+ J K λj K (1 + δK0 ) 12
− 1/2 , −δK−1K λ− J K λj K (1 + δK1 ) 12
(6)
FIG. 1. The Jacobi coordinates for the AB + CD system.
(1D) reference Hamiltonians are defined as 1 ∂2 hˆ i (ri ) = − + V ref (ri ), 2μi ∂ri2
i = 1, 2,
(2)
where V ref (ri ) are the corresponding 1D reference potentials along the coordinates ri . In this work, we use the latest PES for the lowest electronic state of the system,17 despite the experimental evidence showing some non-adiabatic couplings.12 The parity (ε) adapted wave function is expanded in terms of the body-fixed (BF) rovibrational basis functions,25 v1 J Mε r1 , r2 ) = ψ J Mε (R, Cnν v j K un (R)φν (r1 ) 1 2
1
n,ν1 ,v2 ,j,K
ˆ rˆ1 , rˆ2 ), × φν (r2 ) Jj KMε (R, 2
(3)
where n labels the translational basis functions, ν 1 and ν 2 are the vibrational basis indices for r1 and r2 , and the composite index j denotes (j1 , j2 , j12 ). The translational basis funcv tions, un1 , are dependent on ν 1 due to the use of an L-shaped 25 grid. Jj KMε in Eq. (3) are the parity-adapted coupled BF total angular momentum eigenfunctions, which can be written as ∗ j K J Mε −1/2 2J + 1 J DK,M j K = (1 + δK0 ) Yj 12j 1 2 8π j −K J + ε(−1)j1 +j2 +j12 +J D−K,M Yj 12j 1 2 ∗
,
(4)
J where DK,M are the Wigner rotation matrices.26 M is the projection of J on the z axis in the space-fixed (SF) frame and j K K is its projection on the BF z axis that coincides with R. Yj 12j 1 2 are defined as25 YjJ K j1 mj2 K − m | J Kyj m (θ1 , ϕ1 )yj K−m (θ2 , 0), j = 1 2
1
2
m
(5) and yjm denote the spherical harmonics. Note the restriction that ε(−1)j1 +j2 +j12 +J = 0 for K = 0 in Eq. (4). This finite basis representation (FBR) is compact, which has a much smaller size than the corresponding grid. The centrifugal term, i.e., (Jˆ − jˆ12 )2 in the Hamiltonian, which gives rise to the coupling between different K blocks in
1/2 with λ± . Within the coupledAB = [A(A + 1) − B(B + 1)] state or centrifugal-sudden (CS) approximation,27, 28 the couplings between different K blocks are neglected. As a result, the helicity quantum number K becomes conserved. Our implementation of the initial state-selected Chebyshev real wave packet method6, 29, 30 for tetra-atomic systems is well documented in recent publications,31–34 so only a brief outline is provided here. The initial wave packet |χ i is constructed as the direct product of a Gaussian wave packet in the scattering coordinate and specific rovibrational states of both HF and OH in the BF representation,
|χi = N e−(R−R0 ) /2δ cos(ki R)|ν1i |ν2i |j1i j2i j12i Ki , 2
2
(7)
where N is the normalization factor, R0 and δ are the mean position and width of the Gaussian function and ki is the mean momentum given by Ei via ki = 2μR Ei , j1i , j2i , j12i , and Ki are the initial angular quantum numbers. The definition of the wave function in the BF frame requires a large R grid because of the long-range Coriolis coupling.35–37 However, the BF representation is more convenient for the implementation of the CS approximation, where the Coriolis coupling is absent. The wave packet is propagated using the Chebyshev propagator,38–40 |ψk+1 = D(2Hˆ scaled |ψk − D|ψk−1 ),
k ≥ 1,
(8)
where |ψ1 = D Hˆ scaled |ψ0 and |ψ 0 = |χ i . To impose outgoing boundary conditions, the following Gaussian shaped damping function D is applied at the grid edges,
D(x) = e
−α
x
x−x a −x
max
n a
,
(9)
where x = R and r1 , xa is the starting point of the damping function. The scaled Hamiltonian is defined as Hˆ scaled = (Hˆ − H + )/H − to avoid the divergence of the Chebyshev propagator outside the range [−1, 1]. Here the spectral medium and half-width of the Hamiltonian H± = (Hmax ± Hmin )/2 are calculated from the spectral extrema Hmin and Hmax . Since the initial wave packet is real, the propagation can be efficiently and accurately realized in real arithmetic.38, 41 The action of the Hamiltonian onto the FBR wave packet is evaluated with a partial sum.42 To compute the action of the potential energy operator, the FBR wave function is converted to the discrete variable representation (DVR),42 where V is diagonal, via a series of low-dimensional transformations. The coupled angular basis used here25 is more compact than the uncoupled one, but it requires an additional transformation.43
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The flux through the dividing surface, S[r1 = r1F ], is obtained from the energy-dependent scattering wavefunction, ψi+ (E), which is determined by Fourier transforming the
Pi (E) =
1 2π μr |ai (E)|2 (H − )2 sin2 θ 1 (2 − δk0 )e−ikθ ψk ×Im k
∂ (2 − δk 0 )e−ik θ ψ ∂r1 k
(11)
where |φ iE is the free scattering wavefunction. In the CS calculations, the Riccati-Bessel function was used. In the CC calculations, the single K channel Gaussian wave packet was first propagated backward to the reactant asymptote where the two reagents are far from each other and thus the Coriolis coupling is zero. The energy amplitude of the wave packet in the K channel is then calculated, followed by forward propagation to the original position to construct the initial wave packet. The total reaction integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all relevant partial waves, 1 σv v j j (E) = 1 2 1 2 (2j1 + 1)(2j2 + 1) π (2J + 1)PvJ εv j j j K (E) × 2 1 2 1 2 12 k j Kε J ≥K
(10)
,
r1 =r1F
k
where the Chebyshev angle θ = arccos Escaled is a nonlinear mapping of the scaled energy Escaled = (E − H+ )/H− . The energy amplitude of the initial wave packet at the energy E, ai (E) in Eq. (10), is given by ai (E) = φiE | χi ,
wave packet at the dividing surface.44 An energy grid is used for computing the initial state-selected total reaction probability as follows:
propagation, the gradient of the PES was obtained numerically by a central-difference algorithm. The propagation time step was selected to be 0.01 fs, which conserves the energy better than 10−2 kcal/mol for all trajectories in this work. The total ICS is computed according to the following formula: 2 σr (Ec ) = π bmax Pr (Ec ),
(13)
where the reaction probability Pr (Ec ) at the specified collision energy Ec is given by the ratio between the number of reactive trajectories (Nr ) and total number of trajectories (Ntotal ), Pr (Ec ) = Nr /Ntotal , (14) with the standard error given by = (Ntotal −Nr )/Ntotal Nr . III. RESULTS AND DISCUSSION
The numerical parameters employed in the QD calculations are given in Table I. The parameters were extensively tested to give converged results. The propagation requires around 3000 Chebyshev steps. A total of 90 partial waves were needed to converge the ICSs.
12
=
j Kε 1 σ 12 (E), (12) (2j1 + 1)(2j2 + 1) j Kε v1 v2 j1 j2 12
j Kε
where σv 12v j j (E) is the j12 , K, and ε specific cross section 1 2 1 2 and K is taken from 0 to min(j12 , J). B. Quasi-classical trajectory method
The standard QCT method implemented in VENUS45 was also employed for comparison. Batches (105 –106 ) of trajectories were calculated with translational energies of 20–40 kcal/mol to make the statistical errors all within 5%. The initial ro-vibrational energies of HF and OH were sampled using the conventional semi-classical method.46 The maximal impact parameter (bmax ) was determined using small batches of trajectories with trial values for each set of quantum numbers and collision energy. The trajectories were initiated with a reactant separation of 15.0 Å and terminated when products reached a separation of 7.0 Å, or when reactants are separated by 10.0 Å for non-reactive trajectories. During the
A. Accuracy of the CS approximation
Since only one K block is used in the calculation, the CS approximation substantially saves the computational time and memory. The accuracy of the CS approximation has been tested in several reactions in the past47–52 and it has been TABLE I. Numerical parameters used in the wave packet calculations. (Atomic units are used unless stated otherwise.) Grid/basis range and size
R ∈ [2.5, 17.0], NRtot = 220, NRint = 80 asy
Nrint = 41, Nr1 = 5 1
Nr = 5 2
j1max = 58, j2max = 20 Initial wave packet
R0 = 14.5, δ = 0.2, Ei = 1.0 eV
Damping term
Ra = 14.0, αR = 0.05, nR = 2.5, r1a = 4.0,
Flux position
r1F
αr = 0.05, nr = 2.5 1
1
= 4.0
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FIG. 2. Comparison of CS and CC ICSs for the HF(ν 1 = j1 = 0) + OH(ν 2 = j2 = 0) reaction.
found that this approximation generally works well for reactions with a well-defined tight barrier. In this study, we first examine the accuracy of the CS approximation for the title reaction by comparing it with the converged coupled-channel (CC) results, in which the truncated number of the nearest K blocks is taken as min(3, J + 1). The fact that only a few K blocks were needed to converge the CC results indicates that the Coriolis coupling is relatively small. Figure 2 shows the calculated CC and CS ICSs as a function of the translational energy for the title reaction from the ground rovibrational states of both reagents. The CS ICS is slightly smaller than its CC counterpart over the entire energy range. The difference between them increases with the translational energy, with errors no larger than 20%, consistent with previous observations. This difference suggests that there exist some couplings among different K blocks and the Coriolis coupling enhances the reactivity. Since the CS approximation greatly reduces computational costs, the results presented below are exclusively based on this approximation. B. Vibrational effects of HF and OH
Figure 3 shows the QD ICSs for the HF (ν 1 = 0, 1; j1 = 0) + OH (ν 2 = j2 = 0) reaction as a function of the translational energy (upper panel) and total energy (lower panel). The corresponding QCT results are also presented in the figure for comparison. From the upper panel, it can be readily seen that the reaction threshold is shifted to a lower energy when HF is excited vibrationally. In addition, the ICS for ν 1 = 1 rises much more steeply than that for ν 1 = 0. The energy threshold reduction is about 0.62 eV, larger than the HF vibrational gap of 0.49 eV. This is partly due to the HF mode softening as the system approaches the transition state.21 Indeed, the vibrational adiabatic barriers for the ground and vibrational excited HF are 0.83 and 0.64 eV. The additional lowering of the threshold is presumably due to the vibrational energy in the excited HF. In the lower panel, the comparison of the ICSs in the total energy scale clearly indicates that the HF vibrational energy is more efficient than translational energy in promoting the “late”-barrier reaction.
FIG. 3. QD and QCT ICSs for the HF (ν 1 = 0, 1; j1 = 0) + OH (ν 2 = j2 = 0) reaction as a function of the translational energy and total energy.
The QCT ICSs agree well with the corresponding QD results in both the profile and amplitude. However, the QCT cross sections appear to be slightly larger than the QD results over the entire energy range, possibly caused by the ZPE leakage, an intrinsic defect in the QCT methodology.53, 54 On the other hand, it is also possible that the QD results were underestimated with the CS approximation, as discussed above. However, the quantitative difference between the two methods does not affect the conclusion on mode specificity. In Fig. 4, we compare the QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0, 1; j2 = 0) reaction. Both the QD and QCT results indicate that the vibrational excitation of OH has a negligible effect on the reaction. In other words, the hydroxyl radical can be considered as a spectator in this reaction, at least in the energy range studied here. The spectator nature of OH is reminiscent of several other reactions involving the hydroxyl radical.25, 52, 55 Again, the QCT ICSs are larger than their QD counterpart, presumably due to the same reason as discussed above, but the similar lack of enhancement is also observed. C. Rotational effects of HF and OH
Figure 5(a) displays the K-specific and K-averaged QD ICSs for the HF (ν 1 = 0; j1 = 1) + OH (ν 2 = 0; j2 = 0) reaction as a function of the translational energy. The different initial K values represent different relative orientations of
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FIG. 4. QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0, 1; j2 = 0) reaction as a function of the translational energy.
the reagents. Clearly, the ICS for K = 1 is significantly larger than that for K = 0 at the collision energy above the reaction threshold. For j1 = 2, it can be seen from Fig. 5(b) that, as K increases, the cross section first decreases, and then increases. Interestingly, the cross section for K = 2 is close to but somewhat larger than that for K = 0. It appears that for both j1 = 1 and 2 the initial K value has a significant effect on FIG. 6. K-specific and K-averaged QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0; j2 ) reaction. (a) j2 = 1; (b) j2 = 2; and (c) j2 = 0–2.
the amplitude of the cross section but has negligible effect on the energy threshold. In Fig. 5(c), the K-averaged cross sections for the reagent HF in the first two rotationally excited states are plotted as a function of the total energy. Obviously, the ICS increases slightly with the value of j1 . Thus, the rotational excitation of HF enhances the reaction, albeit slightly. It can be seen that the enhancement effect is well reproduced by the QCT results, despite the fact that the QCT cross section is larger than the QD counterpart over the energy range. The K-specific and K-averaged QD ICSs for the OH reagent in the first and second excited rotational states are presented in Figs. 6(a) and 6(b), respectively. For j2 = 1, as K increases, the cross section becomes larger above the threshold. This is similar to the behavior for j1 = 1. For j2 = 2, the K-specific cross section increases with the value of K, which is quite different from the character for j1 = 2. Figure 5(c) shows the K-averaged QD and QCT ICSs for the first two rotationally excited states. Similar to the reagent HF, the rotational excitation of OH also promotes the reaction slightly. Interestingly, the QD cross section is again smaller than the QCT result. D. Sudden vector projection model FIG. 5. K-specific and K-averaged QD and QCT ICSs for the HF(ν 1 = 0; j1 ) + OH (ν 2 = j2 = 0) reaction. (a) j1 = 1; (b) j1 = 2; and (c) j1 = 0–2.
The mode specificity presented in the results above is consistent with the observed product distribution of the
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·Q ) for the HF + OH reaction. TABLE II. SVP values (Pi = Q i RC Species
SVP
Mode
HF
0.99 0.04 0.01 0.05 0.02 0.12
Vibration Rotation Vibration Rotation Translation Relative rotation
OH HF–OH
reverse F + H2 O reaction.8–12 Indeed, the vibrational distribution of the HF product in the F + H2 O reaction was found to be inverted, while OH was found in its ground vibrational state. In addition, their rotational distributions are all cold. This presents a good illustration of the well-known microscopic reversibility at quantum state selected level. The observed mode specificity in the title reaction can be rationalized using our recently proposed SVP model.23, 24, 56, 57 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. The large vibrational frequencies of the two reactants in the title reaction clearly favor such a disparity of the time scales. In the sudden limit, 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 i ) and the rethe overlap between the reactant normal mode (Q action coordinate vector (QRC ): Pi = Qi · QRC ∈ [0, 1]. Apparently, the larger the overlap, the higher capacity for enhancing reactivity. Table II shows the calculated SVP values for the title reaction. It can be seen that the vibrational mode of HF has the largest projection on the reaction coordinate at the transition state, indicating it is essentially the reaction coordinate. The SVP value for the translational mode is quite small, indicating a very weak coupling with the reaction coordinate. Thus, the vibrational excitation of HF is predicted to promote the reaction more effectively than translational energy. This is confirmed by both the QD and QCT results presented above. In addition, the vibrational mode of OH has a negligible SVP value, indicative of very weak coupling with the reaction coordinate. The OH radical thus should be a good spectator in the reaction. As discussed above, our dynamical calculations confirm that the vibrational excitation of OH has a negligible effect on the reaction. These effects of vibrational excitations have already been discussed before in the context of product energy disposal for the F + H2 O reaction, which is the reverse of the title reaction.24 Finally, the SVP values for the diatomic rotations are rather small, suggesting weak coupling of the reactant rotational modes with the reaction coordinate at the transition state. These predictions are consistent with the small enhancement in the reactivity. However, it should be noted that the rotational effects are much harder to predict due to angular momentum coupling.56, 58, 59 We note in passing that Polanyi’s rules can explain the HF vibrational effect in terms of the “late” barrier in the title reaction, but provide no guidance on the rotational effects. The rotational effects
might also be complicated by the pre-reaction well that might exert a significant stereodynamic force on the incoming reactants, as demonstrated in several reactions.20, 60 IV. CONCLUSIONS
The reaction dynamics and mode selectivity of the HF + OH → F + H2 O reaction have been extensively investigated using the initial state-selected Chebyshev real wave packet method and the QCT method. The dynamical calculations were carried out on an accurate ab initio based global PES. It was found that the CS approximation slightly underestimates the cross sections when compared with the CC results over the entire energy range. More interestingly, the vibrational excitation of HF was found in both QD and QCT calculations to significantly enhance the reaction while the vibrational excitation of OH has a negligible effect on the reaction. In addition, the rotational excitation of each reagent promotes the reaction slightly. The mode specificity observed here is consistent with the product state distribution of the reverse F + H2 O reaction and can be readily explained by the SVP model. This work sets the stage for exploring the reactant internal excitation on the product state distributions in this reaction. In addition, it might also be interesting to compare the reaction studied here to the Cl + H2 O ↔ HCl + OH reaction, which has a “late” barrier in the forward direction.61, 62 The mode specificity and possible intramolecular vibrational energy redistribution24, 63–69 should shed further light on these prototypical four atom reactions. ACKNOWLEDGMENTS
This work was supported by the Department of Energy (Grant No. DE-FG02-05ER15694 to H.G.) and calculations were performed at the National Energy Research Scientific Computing (NERSC) Center. 1 D.
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22 J.
48 D.
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H. Zhang and S.-Y. Lee, J. Chem. Phys. 112, 203 (2000). H. Zhang, M. Yang, and S.-Y. Lee, J. Chem. Phys. 117, 10067 (2002). 50 B. Fu, Y. Zhou, and D. H. Zhang, Chem. Sci. 3, 270 (2012). 51 H. Song, Y. Lu, and S.-Y. Lee, J. Chem. Phys. 136, 114307 (2012). 52 H. Song, S.-Y. Lee, M. Yang, and Y. Lu, J. Chem. Phys. 139, 154310 (2013). 53 K. N. Swamy and W. L. Hase, J. Phys. Chem. 87, 4715 (1983). 54 J. M. Bowman, B. Gazdy, and Q. Sun, J. Chem. Phys. 91, 2859 (1989). 55 D. Neuhauser, J. Chem. Phys. 100, 9272 (1994). 56 B. Jiang, J. Li, and H. Guo, J. Chem. Phys. 140, 034112 (2014). 57 J. Li and H. Guo, J. Phys. Chem. A 118, 2419 (2014). 58 R. Liu, F. Wang, B. Jiang, G. Czakó, M. Yang, K. Liu, and H. Guo, J. Chem. Phys. 141, 074310 (2014). 59 H. Song, J. Li, B. Jiang, M. Yang, Y. Lu, and H. Guo, J. Chem. Phys. 140, 084307 (2014). 60 D. Skouteris, D. E. Manolopoulos, W. Bian, H.-J. Werner, L.-H. Lai, and K. Liu, Science 286, 1713 (1999). 61 Y. Guo, M. Zhang, Y. Xie, and H. F. Schaefer III, J. Chem. Phys. 139, 041101 (2013). 62 J. Li, R. Dawes, and H. Guo, J. Chem. Phys. 139, 074302 (2013). 63 A. Sinha, J. D. Thoemke, and F. F. Crim, J. Chem. Phys. 96, 372 (1992). 64 G. Nyman and D. C. Clary, J. Chem. Phys. 100, 3556 (1994). 65 K. Kudla and G. C. Schatz, Chem. Phys. 175, 71 (1993). 66 D. C. Clary, G. Nyman, and R. Hernandez, J. Chem. Phys. 101, 3704 (1994). 67 J. D. Thoemke, J. M. Pfeiffer, R. B. Metz, and F. F. Crim, J. Phys. Chem. 99, 13748 (1995). 68 H. G. Yu and G. Nyman, J. Chem. Phys. 113, 8936 (2000). 69 J. R. Fair, D. Schaefer, R. Kosloff, and D. J. Nesbitt, J. Chem. Phys. 116, 1406 (2002).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 164316 (2014)
Mode specificity in the HF + OH → F + H2 O reaction Hongwei Song, Jun Li, and Hua Guoa) Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA
(Received 29 August 2014; accepted 15 October 2014; published online 28 October 2014) Full-dimensional quantum dynamics and quasi-classical trajectory calculations are reported for the title reaction on a recently constructed ab initio based global potential energy surface. Strong mode specificity was found, consistent with the prediction of the sudden vector projection model. Specifically, the HF vibration strongly promotes the reaction while the OH vibration has little effect. Rotational excitations of both reactants slightly enhance the reaction. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4900445] I. INTRODUCTION
Reactions involving four atoms present an ideal proving ground to study reaction dynamics.1–3 This is because such systems offer much richer complexity than atom-diatom reactions but are sufficiently small to be amenable to fulldimensional quantum dynamical characterization on accurate global potential energy surfaces (PESs).4–6 With six degrees of freedom, mode specificity in these systems provides valuable insights into reaction dynamics.7 In this work, we focus on the F + H2 O → HF + OH reaction. Experimentally, the products have been found to be dominated by strong HF vibrational excitation while OH has essentially no internal excitation, behaving as a spectator.8–12 Evidence is also presented for non-adiabatic transitions between the lowest two electronic states.12 Theoretically, ab initio studies revealed that the reaction path on the lowest electronic state features a low and “early” barrier, flanked by significant pre- and post-transition state wells.13, 14 Several full-dimensional global PESs have been developed based on large numbers of high-level ab initio points,15, 16 cumulating with an externally scaled PES that has the correct exothermicity and barrier height17 and is capable of reproducing the measured cross section.18 Quantum dynamics (QD) and quasi-classical trajectory (QCT) studies on these PESs have revealed many surprising and interesting behaviors. For instance, it was found that the hemi-bonded pre-transition state well19 significantly enhances the reactivity at low collision energies by steering the reactants towards the barrier.20 Furthermore, excitations of all vibrational modes of H2 O were found to promote the reaction more effectively than translational energy.21 The mode specificity was surprising as the reactant vibrational modes were expected to have lower efficacies than translation for this “early” barrier reaction, according to Polanyi’s Rules.22 Furthermore, rotational excitations of the H2 O reactant were also shown to enhance reactivity substantially.18 These vibrational and rotational mode specific effects can nonetheless be explained by the recently proposed Sudden Vector Projection (SVP) model,23 which ata) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/141(16)/164316/7/$30.00
tributes the mode specificity to the coupling between the reactant modes and the reaction coordinate at the transition state.24 In this work, we investigate the HF + OH → F + H2 O reaction, which is the reverse of the much studied F + H2 O reaction. While this endothermic reaction has been investigated neither experimentally nor theoretically, it presents a prototype to validate the SVP model for diatom-diatom reactions. In particular, we examine in this work effects of reactant vibrational and rotational excitations on the overall reactivity using both QD and QCT methods on an accurate PES. We note in particular that unlike the state-to-state S-matrix elements, the initial state specific cross sections reported here cannot be directly obtained from detailed balance, although microscopic reversibility does provide some qualitative insights for reactivities in both directions of the reaction. This article is organized as follows. Section II details the theoretical methods used to investigate the reaction dynamics and mode specificity. The results are presented and discussed in Sec. III. Finally, conclusions are given in Sec. IV. II. THEORY A. Quantum dynamics
The full-dimensional Hamiltonian for the diatom-diatom reaction in the reactant Jacobi coordinates, as shown in Fig. 1, for a given total angular momentum J can be written as (¯ = 1 thereafter)25 ˆ ˆ 2 1 ∂2 ˆ 1 (r1 ) + hˆ 2 (r2 ) + (J − j12 ) Hˆ = − + h 2μR ∂R 2 2μR R 2 +
jˆ12 jˆ22 + + Vˆ (R, r1 , r2 , θ1 , θ2 , ϕ1 ) 2μ1 r12 2μ2 r22 ref
ref
− V1 (r1 ) − V2 (r2 ),
(1)
where r1 , r2 , and R are the bond distances of AB and CD, and the distance between their centers of mass (COMs), respectively, with μ1 , μ2 , and μR as their corresponding reduced masses. jˆ1 and jˆ2 are the rotational angular momentum operators of AB and CD, which are coupled to jˆ12 . Jˆ denotes the total angular momentum of the system. The one-dimensional
141, 164316-1
© 2014 AIP Publishing LLC
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the BF representation, is given by J Mε j K (Jˆ − jˆ12 )2 Jj Mε K = δjj δKK [J (J + 1) +j12 (j12 + 1) − 2K 2 ] + 1/2 −δK+1K λ+ J K λj K (1 + δK0 ) 12
− 1/2 , −δK−1K λ− J K λj K (1 + δK1 ) 12
(6)
FIG. 1. The Jacobi coordinates for the AB + CD system.
(1D) reference Hamiltonians are defined as 1 ∂2 hˆ i (ri ) = − + V ref (ri ), 2μi ∂ri2
i = 1, 2,
(2)
where V ref (ri ) are the corresponding 1D reference potentials along the coordinates ri . In this work, we use the latest PES for the lowest electronic state of the system,17 despite the experimental evidence showing some non-adiabatic couplings.12 The parity (ε) adapted wave function is expanded in terms of the body-fixed (BF) rovibrational basis functions,25 v1 J Mε r1 , r2 ) = ψ J Mε (R, Cnν v j K un (R)φν (r1 ) 1 2
1
n,ν1 ,v2 ,j,K
ˆ rˆ1 , rˆ2 ), × φν (r2 ) Jj KMε (R, 2
(3)
where n labels the translational basis functions, ν 1 and ν 2 are the vibrational basis indices for r1 and r2 , and the composite index j denotes (j1 , j2 , j12 ). The translational basis funcv tions, un1 , are dependent on ν 1 due to the use of an L-shaped 25 grid. Jj KMε in Eq. (3) are the parity-adapted coupled BF total angular momentum eigenfunctions, which can be written as ∗ j K J Mε −1/2 2J + 1 J DK,M j K = (1 + δK0 ) Yj 12j 1 2 8π j −K J + ε(−1)j1 +j2 +j12 +J D−K,M Yj 12j 1 2 ∗
,
(4)
J where DK,M are the Wigner rotation matrices.26 M is the projection of J on the z axis in the space-fixed (SF) frame and j K K is its projection on the BF z axis that coincides with R. Yj 12j 1 2 are defined as25 YjJ K j1 mj2 K − m | J Kyj m (θ1 , ϕ1 )yj K−m (θ2 , 0), j = 1 2
1
2
m
(5) and yjm denote the spherical harmonics. Note the restriction that ε(−1)j1 +j2 +j12 +J = 0 for K = 0 in Eq. (4). This finite basis representation (FBR) is compact, which has a much smaller size than the corresponding grid. The centrifugal term, i.e., (Jˆ − jˆ12 )2 in the Hamiltonian, which gives rise to the coupling between different K blocks in
1/2 with λ± . Within the coupledAB = [A(A + 1) − B(B + 1)] state or centrifugal-sudden (CS) approximation,27, 28 the couplings between different K blocks are neglected. As a result, the helicity quantum number K becomes conserved. Our implementation of the initial state-selected Chebyshev real wave packet method6, 29, 30 for tetra-atomic systems is well documented in recent publications,31–34 so only a brief outline is provided here. The initial wave packet |χ i is constructed as the direct product of a Gaussian wave packet in the scattering coordinate and specific rovibrational states of both HF and OH in the BF representation,
|χi = N e−(R−R0 ) /2δ cos(ki R)|ν1i |ν2i |j1i j2i j12i Ki , 2
2
(7)
where N is the normalization factor, R0 and δ are the mean position and width of the Gaussian function and ki is the mean momentum given by Ei via ki = 2μR Ei , j1i , j2i , j12i , and Ki are the initial angular quantum numbers. The definition of the wave function in the BF frame requires a large R grid because of the long-range Coriolis coupling.35–37 However, the BF representation is more convenient for the implementation of the CS approximation, where the Coriolis coupling is absent. The wave packet is propagated using the Chebyshev propagator,38–40 |ψk+1 = D(2Hˆ scaled |ψk − D|ψk−1 ),
k ≥ 1,
(8)
where |ψ1 = D Hˆ scaled |ψ0 and |ψ 0 = |χ i . To impose outgoing boundary conditions, the following Gaussian shaped damping function D is applied at the grid edges,
D(x) = e
−α
x
x−x a −x
max
n a
,
(9)
where x = R and r1 , xa is the starting point of the damping function. The scaled Hamiltonian is defined as Hˆ scaled = (Hˆ − H + )/H − to avoid the divergence of the Chebyshev propagator outside the range [−1, 1]. Here the spectral medium and half-width of the Hamiltonian H± = (Hmax ± Hmin )/2 are calculated from the spectral extrema Hmin and Hmax . Since the initial wave packet is real, the propagation can be efficiently and accurately realized in real arithmetic.38, 41 The action of the Hamiltonian onto the FBR wave packet is evaluated with a partial sum.42 To compute the action of the potential energy operator, the FBR wave function is converted to the discrete variable representation (DVR),42 where V is diagonal, via a series of low-dimensional transformations. The coupled angular basis used here25 is more compact than the uncoupled one, but it requires an additional transformation.43
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The flux through the dividing surface, S[r1 = r1F ], is obtained from the energy-dependent scattering wavefunction, ψi+ (E), which is determined by Fourier transforming the
Pi (E) =
1 2π μr |ai (E)|2 (H − )2 sin2 θ 1 (2 − δk0 )e−ikθ ψk ×Im k
∂ (2 − δk 0 )e−ik θ ψ ∂r1 k
(11)
where |φ iE is the free scattering wavefunction. In the CS calculations, the Riccati-Bessel function was used. In the CC calculations, the single K channel Gaussian wave packet was first propagated backward to the reactant asymptote where the two reagents are far from each other and thus the Coriolis coupling is zero. The energy amplitude of the wave packet in the K channel is then calculated, followed by forward propagation to the original position to construct the initial wave packet. The total reaction integral cross section (ICS) from a specific initial state is calculated by summing the reaction probabilities over all relevant partial waves, 1 σv v j j (E) = 1 2 1 2 (2j1 + 1)(2j2 + 1) π (2J + 1)PvJ εv j j j K (E) × 2 1 2 1 2 12 k j Kε J ≥K
(10)
,
r1 =r1F
k
where the Chebyshev angle θ = arccos Escaled is a nonlinear mapping of the scaled energy Escaled = (E − H+ )/H− . The energy amplitude of the initial wave packet at the energy E, ai (E) in Eq. (10), is given by ai (E) = φiE | χi ,
wave packet at the dividing surface.44 An energy grid is used for computing the initial state-selected total reaction probability as follows:
propagation, the gradient of the PES was obtained numerically by a central-difference algorithm. The propagation time step was selected to be 0.01 fs, which conserves the energy better than 10−2 kcal/mol for all trajectories in this work. The total ICS is computed according to the following formula: 2 σr (Ec ) = π bmax Pr (Ec ),
(13)
where the reaction probability Pr (Ec ) at the specified collision energy Ec is given by the ratio between the number of reactive trajectories (Nr ) and total number of trajectories (Ntotal ), Pr (Ec ) = Nr /Ntotal , (14) with the standard error given by = (Ntotal −Nr )/Ntotal Nr . III. RESULTS AND DISCUSSION
The numerical parameters employed in the QD calculations are given in Table I. The parameters were extensively tested to give converged results. The propagation requires around 3000 Chebyshev steps. A total of 90 partial waves were needed to converge the ICSs.
12
=
j Kε 1 σ 12 (E), (12) (2j1 + 1)(2j2 + 1) j Kε v1 v2 j1 j2 12
j Kε
where σv 12v j j (E) is the j12 , K, and ε specific cross section 1 2 1 2 and K is taken from 0 to min(j12 , J). B. Quasi-classical trajectory method
The standard QCT method implemented in VENUS45 was also employed for comparison. Batches (105 –106 ) of trajectories were calculated with translational energies of 20–40 kcal/mol to make the statistical errors all within 5%. The initial ro-vibrational energies of HF and OH were sampled using the conventional semi-classical method.46 The maximal impact parameter (bmax ) was determined using small batches of trajectories with trial values for each set of quantum numbers and collision energy. The trajectories were initiated with a reactant separation of 15.0 Å and terminated when products reached a separation of 7.0 Å, or when reactants are separated by 10.0 Å for non-reactive trajectories. During the
A. Accuracy of the CS approximation
Since only one K block is used in the calculation, the CS approximation substantially saves the computational time and memory. The accuracy of the CS approximation has been tested in several reactions in the past47–52 and it has been TABLE I. Numerical parameters used in the wave packet calculations. (Atomic units are used unless stated otherwise.) Grid/basis range and size
R ∈ [2.5, 17.0], NRtot = 220, NRint = 80 asy
Nrint = 41, Nr1 = 5 1
Nr = 5 2
j1max = 58, j2max = 20 Initial wave packet
R0 = 14.5, δ = 0.2, Ei = 1.0 eV
Damping term
Ra = 14.0, αR = 0.05, nR = 2.5, r1a = 4.0,
Flux position
r1F
αr = 0.05, nr = 2.5 1
1
= 4.0
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FIG. 2. Comparison of CS and CC ICSs for the HF(ν 1 = j1 = 0) + OH(ν 2 = j2 = 0) reaction.
found that this approximation generally works well for reactions with a well-defined tight barrier. In this study, we first examine the accuracy of the CS approximation for the title reaction by comparing it with the converged coupled-channel (CC) results, in which the truncated number of the nearest K blocks is taken as min(3, J + 1). The fact that only a few K blocks were needed to converge the CC results indicates that the Coriolis coupling is relatively small. Figure 2 shows the calculated CC and CS ICSs as a function of the translational energy for the title reaction from the ground rovibrational states of both reagents. The CS ICS is slightly smaller than its CC counterpart over the entire energy range. The difference between them increases with the translational energy, with errors no larger than 20%, consistent with previous observations. This difference suggests that there exist some couplings among different K blocks and the Coriolis coupling enhances the reactivity. Since the CS approximation greatly reduces computational costs, the results presented below are exclusively based on this approximation. B. Vibrational effects of HF and OH
Figure 3 shows the QD ICSs for the HF (ν 1 = 0, 1; j1 = 0) + OH (ν 2 = j2 = 0) reaction as a function of the translational energy (upper panel) and total energy (lower panel). The corresponding QCT results are also presented in the figure for comparison. From the upper panel, it can be readily seen that the reaction threshold is shifted to a lower energy when HF is excited vibrationally. In addition, the ICS for ν 1 = 1 rises much more steeply than that for ν 1 = 0. The energy threshold reduction is about 0.62 eV, larger than the HF vibrational gap of 0.49 eV. This is partly due to the HF mode softening as the system approaches the transition state.21 Indeed, the vibrational adiabatic barriers for the ground and vibrational excited HF are 0.83 and 0.64 eV. The additional lowering of the threshold is presumably due to the vibrational energy in the excited HF. In the lower panel, the comparison of the ICSs in the total energy scale clearly indicates that the HF vibrational energy is more efficient than translational energy in promoting the “late”-barrier reaction.
FIG. 3. QD and QCT ICSs for the HF (ν 1 = 0, 1; j1 = 0) + OH (ν 2 = j2 = 0) reaction as a function of the translational energy and total energy.
The QCT ICSs agree well with the corresponding QD results in both the profile and amplitude. However, the QCT cross sections appear to be slightly larger than the QD results over the entire energy range, possibly caused by the ZPE leakage, an intrinsic defect in the QCT methodology.53, 54 On the other hand, it is also possible that the QD results were underestimated with the CS approximation, as discussed above. However, the quantitative difference between the two methods does not affect the conclusion on mode specificity. In Fig. 4, we compare the QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0, 1; j2 = 0) reaction. Both the QD and QCT results indicate that the vibrational excitation of OH has a negligible effect on the reaction. In other words, the hydroxyl radical can be considered as a spectator in this reaction, at least in the energy range studied here. The spectator nature of OH is reminiscent of several other reactions involving the hydroxyl radical.25, 52, 55 Again, the QCT ICSs are larger than their QD counterpart, presumably due to the same reason as discussed above, but the similar lack of enhancement is also observed. C. Rotational effects of HF and OH
Figure 5(a) displays the K-specific and K-averaged QD ICSs for the HF (ν 1 = 0; j1 = 1) + OH (ν 2 = 0; j2 = 0) reaction as a function of the translational energy. The different initial K values represent different relative orientations of
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J. Chem. Phys. 141, 164316 (2014)
FIG. 4. QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0, 1; j2 = 0) reaction as a function of the translational energy.
the reagents. Clearly, the ICS for K = 1 is significantly larger than that for K = 0 at the collision energy above the reaction threshold. For j1 = 2, it can be seen from Fig. 5(b) that, as K increases, the cross section first decreases, and then increases. Interestingly, the cross section for K = 2 is close to but somewhat larger than that for K = 0. It appears that for both j1 = 1 and 2 the initial K value has a significant effect on FIG. 6. K-specific and K-averaged QD and QCT ICSs for the HF (ν 1 = j1 = 0) + OH (ν 2 = 0; j2 ) reaction. (a) j2 = 1; (b) j2 = 2; and (c) j2 = 0–2.
the amplitude of the cross section but has negligible effect on the energy threshold. In Fig. 5(c), the K-averaged cross sections for the reagent HF in the first two rotationally excited states are plotted as a function of the total energy. Obviously, the ICS increases slightly with the value of j1 . Thus, the rotational excitation of HF enhances the reaction, albeit slightly. It can be seen that the enhancement effect is well reproduced by the QCT results, despite the fact that the QCT cross section is larger than the QD counterpart over the energy range. The K-specific and K-averaged QD ICSs for the OH reagent in the first and second excited rotational states are presented in Figs. 6(a) and 6(b), respectively. For j2 = 1, as K increases, the cross section becomes larger above the threshold. This is similar to the behavior for j1 = 1. For j2 = 2, the K-specific cross section increases with the value of K, which is quite different from the character for j1 = 2. Figure 5(c) shows the K-averaged QD and QCT ICSs for the first two rotationally excited states. Similar to the reagent HF, the rotational excitation of OH also promotes the reaction slightly. Interestingly, the QD cross section is again smaller than the QCT result. D. Sudden vector projection model FIG. 5. K-specific and K-averaged QD and QCT ICSs for the HF(ν 1 = 0; j1 ) + OH (ν 2 = j2 = 0) reaction. (a) j1 = 1; (b) j1 = 2; and (c) j1 = 0–2.
The mode specificity presented in the results above is consistent with the observed product distribution of the
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·Q ) for the HF + OH reaction. TABLE II. SVP values (Pi = Q i RC Species
SVP
Mode
HF
0.99 0.04 0.01 0.05 0.02 0.12
Vibration Rotation Vibration Rotation Translation Relative rotation
OH HF–OH
reverse F + H2 O reaction.8–12 Indeed, the vibrational distribution of the HF product in the F + H2 O reaction was found to be inverted, while OH was found in its ground vibrational state. In addition, their rotational distributions are all cold. This presents a good illustration of the well-known microscopic reversibility at quantum state selected level. The observed mode specificity in the title reaction can be rationalized using our recently proposed SVP model.23, 24, 56, 57 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. The large vibrational frequencies of the two reactants in the title reaction clearly favor such a disparity of the time scales. In the sudden limit, 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 i ) and the rethe overlap between the reactant normal mode (Q action coordinate vector (QRC ): Pi = Qi · QRC ∈ [0, 1]. Apparently, the larger the overlap, the higher capacity for enhancing reactivity. Table II shows the calculated SVP values for the title reaction. It can be seen that the vibrational mode of HF has the largest projection on the reaction coordinate at the transition state, indicating it is essentially the reaction coordinate. The SVP value for the translational mode is quite small, indicating a very weak coupling with the reaction coordinate. Thus, the vibrational excitation of HF is predicted to promote the reaction more effectively than translational energy. This is confirmed by both the QD and QCT results presented above. In addition, the vibrational mode of OH has a negligible SVP value, indicative of very weak coupling with the reaction coordinate. The OH radical thus should be a good spectator in the reaction. As discussed above, our dynamical calculations confirm that the vibrational excitation of OH has a negligible effect on the reaction. These effects of vibrational excitations have already been discussed before in the context of product energy disposal for the F + H2 O reaction, which is the reverse of the title reaction.24 Finally, the SVP values for the diatomic rotations are rather small, suggesting weak coupling of the reactant rotational modes with the reaction coordinate at the transition state. These predictions are consistent with the small enhancement in the reactivity. However, it should be noted that the rotational effects are much harder to predict due to angular momentum coupling.56, 58, 59 We note in passing that Polanyi’s rules can explain the HF vibrational effect in terms of the “late” barrier in the title reaction, but provide no guidance on the rotational effects. The rotational effects
might also be complicated by the pre-reaction well that might exert a significant stereodynamic force on the incoming reactants, as demonstrated in several reactions.20, 60 IV. CONCLUSIONS
The reaction dynamics and mode selectivity of the HF + OH → F + H2 O reaction have been extensively investigated using the initial state-selected Chebyshev real wave packet method and the QCT method. The dynamical calculations were carried out on an accurate ab initio based global PES. It was found that the CS approximation slightly underestimates the cross sections when compared with the CC results over the entire energy range. More interestingly, the vibrational excitation of HF was found in both QD and QCT calculations to significantly enhance the reaction while the vibrational excitation of OH has a negligible effect on the reaction. In addition, the rotational excitation of each reagent promotes the reaction slightly. The mode specificity observed here is consistent with the product state distribution of the reverse F + H2 O reaction and can be readily explained by the SVP model. This work sets the stage for exploring the reactant internal excitation on the product state distributions in this reaction. In addition, it might also be interesting to compare the reaction studied here to the Cl + H2 O ↔ HCl + OH reaction, which has a “late” barrier in the forward direction.61, 62 The mode specificity and possible intramolecular vibrational energy redistribution24, 63–69 should shed further light on these prototypical four atom reactions. ACKNOWLEDGMENTS
This work was supported by the Department of Energy (Grant No. DE-FG02-05ER15694 to H.G.) and calculations were performed at the National Energy Research Scientific Computing (NERSC) Center. 1 D.
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