Quasiclassical Trajectory Calculations of the Rate Constant of the OH

Letter pubs.acs.org/JPCL

Quasiclassical Trajectory Calculations of the Rate Constant of the OH + HBr → Br + H2O Reaction Using a Full-Dimensional Ab Initio Potential Energy Surface Over the Temperature Range 5 to 500 K Antonio G. S. de Oliveira-Filho,†,‡ Fernando R. Ornellas,† and Joel M. Bowman*,‡ †

Departamento de Química Fundamental, Instituto de Química, Universidade de São Paulo, São Paulo, São Paulo, 05508-000, Brazil Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory University, Atlanta, Georgia 30322, United States

‡

S Supporting Information *

ABSTRACT: We report a permutationally invariant, ab initio potential energy surface (PES) for the OH + HBr → Br + H2O reaction. The PES is a fit to roughly 26 000 spin-free UCCSD(T)/cc-pVDZ-F12a energies and has no classical barrier to reaction. It is used in quasiclassical trajectory calculations with a focus on the thermal rate constant, k(T), over the temperature range 5 to 500 K. Comparisons with available experimental data over the temperature range 23 to 416 K are made using three approaches to treat the OH rotational and associated electronic partition function. All display an inverse temperature dependence of k(T) below roughly 160 K and a nearly constant temperature dependence above 160 K, in agreement with experiment. The calculated rate constant with no treatment of spin− orbit coupling is overall in the best agreement with experiment, being (probably fortuitously) within 20% of it. SECTION: Kinetics and Dynamics he OH + HBr → H2O + Br reaction is the major process by which atomic bromine, an active ozone depletion catalyst, is regenerated from the relatively stable hydrogen bromide.1,2 Thermal rate constants for this reaction have been measured over a broad temperature range (23−416 K), and revealed a strong inverse temperature dependence of the rate constant below 150 K.3−14 This surprising behavior of rapid reactivity at very low temperatures was observed, using the CRESU (Cinétique de Réaction en Encoulement Supersonique Uniforme) method.9,10 This method also has been used to study many reactions between neutral species,15,16 making this class of reactions important for the understanding of the chemistry of low-temperature molecular clouds in the interstellar medium. From a fundamental perspective, these reactions pose challenges for the understanding of the underlying dynamics responsible for this behavior. Several approaches have been used in previous theoretical work on the OH + HBr reaction,17−22 but none of these studies conclusively address the issue of the barrier height, and the role of long-range interactions and tunneling in the observed reaction rates. Clary et al.17 used a rotationally adiabatic capture theory with a long-range electrostatic interaction potential to predict a maximum for the rate constant at the temperature of 20 K, a feature that was not observed experimentally, perhaps due to the absence of measurements in the very lowtemperature limit (T < 10 K). Later, Clary et al.18 developed a London−Eyring−Polanyi−Sato (LEPS) potential that was used in a reduced dimensionality quantum scattering

T

© 2014 American Chemical Society

calculations. Based on the dependence of the reaction cross section on the OH rotational quantum number, they obtained a rate constant with a T−1/2 dependence, in accord with the earlier capture model. However, agreement with available experimental data was not good, with theory below experiment and experiment not showing the maximum found in the calculations. This potential, which has no potential barrier, was also used in trajectory calculations, conducted by Nizamov et al.,19 with a focus on experiments that determined the H2O vibrational energy distribution. In the most recent calculations, Liu et al.21 reported a direct ab initio approach to this reaction, employing reaction-path dual-level direct dynamics variational transition-state theory, and attributed the observed lowtemperature inverse T-dependence of the rate constant to tunneling through a small barrier. As noted in the Abstract, the present results find no barrier, and the disagreement with these most recent ones are focused on below. In this Letter, we report a full-dimensional potential energy surface (PES) for the title reaction. This is a permutationally invariant fit to roughly 26 000 high-level ab initio energies. We also report an accurate determination of the energetics of the stationary points of products of this reaction. Quasiclassical trajectories (QCT) were used to calculate the rate constant Received: January 6, 2014 Accepted: February 3, 2014 Published: February 3, 2014 706

dx.doi.org/10.1021/jz5000325 | J. Phys. Chem. Lett. 2014, 5, 706−712

The Journal of Physical Chemistry Letters

Letter

over the temperature range of 5−500 K. The present study also relates to recent investigations of the X + H2O → HX + OH (X = F and Cl) forward and reverse reactions,23−31 for which ab initio potential energy surfaces have been reported. Note in these cases there are reaction barriers for the diatom−diatom reaction, in contrast to the present case. This series of systems contributes greatly to the general understanding of chemical reactivity due to the larger range of reaction dynamics, compared to atom−diatom reaction, that can be studied. The PES is a fit to electronic energies calculated using spinunrestricted, explicitly correlated coupled-cluster 32−34 (UCCSD(T)-F12a) theory with restricted open-shell Hartree−Fock (ROHF) reference functions. These calculations employed the cc-pVDZ-F1235−37 orbital basis sets, together with the following choice of auxiliary basis sets: the cc-pVDZ/ OptRI38 sets were used for the resolution of the identity, the ccpVTZ/JKFIT (for H and O) and QZVPP/JKFIT39,40 (for Br) sets were used for the density fitting of the exchange and Fock operators, while the aug-cc-pVTZ/MP2FIT39 (for H and O) and cc-pVTZ-F12/MP2FIT (for Br) sets were used for the remaining two-electron integrals. In all calculations, except when otherwise indicated, the inner core of bromine was replaced by a relativistic pseudopotential.41 The full-dimensional potential energy surface was constructed using the invariant polynomial method via monomial symmetrization.42,43 The expression for the potential, in this approach, is given by V (y1 , ..., y6 ) =

∑

Figure 1. Schematic reaction path and stationary points for the OH + HBr → H2O + Br reaction. Energies in kcal mol−1.

Cn1...n6y1n1 y6n6 [y2n2 y3n3 y4n4 y5n5 + y2n4 y3n5 y4n2 y5n3 ]

n1,..., n6

(1)

where yi = e−ri/α with α = 2a0, and ri is an internuclear distance, namely: r1 = rHH′, r2 = rHO, r3 = rHBr, r4 = rH′O, r5 = rH′Br and r6 = rOBr. The total order of polynomials involved in the expression (N = n1 + n2 + n3 + n4 + n5 + n6) does not exceed six, resulting in a total of 502 terms. The expansion coefficients in eq 1 were determined by a standard least-squares fit of 26 121 electronic energies. These configurations were first sampled using grids around the stationary points on the PES that were used to fit a tentative potential, on which batches of exploratory trajectories were calculated, at several collision energies. To ensure an accurate description of the long-range behavior, the data set extended to R values as long as 8.5 Å, where R is the distance between the centers of mass of the reactants. Configurations in regions with unphysical behavior were added to the data set, that was then fitted, providing a new tentative potential. This procedure was repeated in order to ensure that all relevant features for the description of the OH + HBr → H2O + Br were properly represented. The overall root-mean-square error (RMSE), that includes points as high as 250 kcal mol−1 above the reactants, is 0.84 kcal mol−1. Restricting the energy threshold to points up to 50 kcal mol−1 above the reactants reduces RMSE to 0.52 kcal mol−1. A schematic reaction path and the energetics of the stationary points are shown in Figure 1. As seen, there is a van der Waals well (one of two), followed by a negative energy saddle-point barrier, both in the entrance channel, followed by a van der Waals well in the product channel. A contour plot in Figure 2 displays a relaxed equipotential contour plot in which the energy, in kcal mol−1 and relative to the OH(re) + HBr(r′e ) asymptotic limit, is shown as a function of the rOH and rHBr distances, for optimized values of the other geometrical parameters. As seen, the potential is smooth and the key

Figure 2. Equipotential contour plot of the H2OBr PES in (rOH′, rH′Br) space with all other internal coordinates optimized. The zero of energy is the OH(re) + HBr(re) asymptotic limit, and the equipotential lines are separated by 5 kcal mol−1.

structures (van der Waals minima and the saddle point) on the reaction path are easily identified. From the perspective of the dynamics, this reaction profile, with a small negative-energy barrier, and large exoergicity suggests that tunneling plays a minor or no role in the dynamics. Thus, quasiclassical trajectory calculations should be reliable for both the rate constant and final-state properties of the products. Here the emphasis is on the rate constant, which as noted above, displays an interesting inverse temperature dependence at low temperatures and is nearly constant, at least up to roughly 500 K. Complementing these calculations, optimized geometries and harmonic frequencies were determined for the stationary points at the CCSD(T)-F12a/cc-pVnZ-F12 (n = D and T) levels of theory, and also using conventional coupled cluster with single, double, and perturbative estimate of connected triples, 44,45 CCSD(T), employing the aug-cc-pVnZ(PP)41,46−48 (n = T and Q) basis sets. Finally, in order to obtain accurate benchmark energetics, a series of single-point calculations at the optimized CCSD(T)F12a/aVTZ-F12 geometries was carried out for the stationary points of the PES, using a procedure similar to the one used to 707



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Table 1. Energies (in kcal mol−1) and Geometries (in Ångström and Degrees) of the Stationary Points on the H2OBr PES

calculate an accurate barrier height of the O + HBr reaction.49−51 The complete basis set (CBS) limit was estimated using a 1/l3max extrapolation52 of CCSD(T) energies calculated with the aug-cc-pVQZ(-PP) and the aug-cc-pV5Z(PP) basis sets.41,46−48 Additive corrections to include the effect of full iterative triple and quadruple excitations, core−valence correlation, scalar relativity and spin−orbit have been calculated. The triple excitation correction (ΔT) was defined as the difference between CCSD(T) and CCSDT53−55 calculations using the cc-pVTZ(-PP) basis set; similarly, the quadruple excitation correction (ΔQ) was calculated from CCSDT and CCSDTQ56−58 energies using the cc-pVDZ(-PP) basis set. The core−valence correlation was included at the CCSD(T) level using the aug-cc-pwCVQZ(-PP) basis set59,60 as the difference between frozen-core calculations and those with the O 1s and Br 3s3p3d electrons correlated (both in the same basis set). The second-order Douglas−Kroll−Hess (DKH) Hamiltonian61 along with the DKH-contracted basis sets,62,63 aug-cc-pVQZ-DK, was used to include scalar relativistic effects on H and O, as well as to also correct for the pseudopotential approximation. This contribution was defined as the difference between CCSD(T)/aug-cc-pVQZ(PP) and DK-CCSD(T)/aug-cc-pVQZ-DK energies within the frozen-core approximation. The inclusion of spin−orbit effects was calculated using the full Breit−Pauli operator within the state interacting approach64 using dynamically weighted state averaged complete active space self-consistent field65−67 (DWSA-CASSCF) wave functions using the aug-cc-pVTZ(-PP) basis sets in a calculation for five states, with a β−1 factor of 3 eV, employing valence active space. This additive correction was defined as the difference between nonspin−orbit and spin− orbit energies. All CCSD(T), CCSD(T)-F12a, and CASSCF calculations were carried out with the Molpro68,69 suite of programs, while CCSDT and CCSDTQ calculations were performed with the MRCC70 as interfaced to Molpro. Table 1 presents CCSD(T) energies and geometries of the relevant indicated stationary points using a variety of basis sets, along with results from the PES. As seen the results from the various basis sets all agree well and the PES reproduces the VDZ-F12 results, upon which it is based, very well. In several instances the PES is fortuitously closer to the “TZ” basis results than the VDZ-F12 ones. Focusing in particular on the saddle point in the entrance channel, we see that all results give a negative energy for this. This is one example where the PES energy of −0.52 kcal/mol is quite close to the “TZ” basis results. A comparison of corresponding harmonic frequencies is given in Supporting Information (SI), where the PES is shown to be generally in good agreement with the direct ab initio results. The frequencies shown there are substantially different from the ones reported previously for LEPS potential18 and also the lower-level direct ab initio calculations.21 Note that harmonic frequencies are not employed in a direct way in the present quasiclassical trajectory calculations. An examination of larger basis sets and extrapolated complete basis set results, spin−orbit interaction, scalar relativistic, core−valence effects, and full iterative triple and quadruple excitations is given in SI. Based on these, the best estimate for the saddle point energy is −0.61 kcal mol−1, which is 0.09 kcal mol−1 below the PES value. As expected spin−orbit coupling is largest in the region of the products, Br···H2O and the products themselves. The best estimate for the product energy is −35.43 kcal mol−1, which, as expected, is lower than

θHOH′

θOH′Br

ϕHOH′Br

−2.96 −2.91 −2.83

1.415 R-vdW (HO···HBr) 0.975 2.148 1.426 112.6 0.972 2.127 1.426 112.9 0.972 2.132 1.424 113.3

173.1 173.6 173.7

0.0 0.0 0.0

−2.85

0.972

112.8

173.5

0.0

−3.19

0.972

173.1

0.0

−2.13 −1.98 −1.67

2.316 1.421 116.5 R′-vdW (HBr···HO) 0.976 3.804 1.421 29.2 0.973 3.778 1.420 31.0 0.973 4.128 1.418 18.6

71.7 73.2 62.0

0.0 0.0 0.0

−1.67

0.973

20.3

65.9

0.0

−1.99

0.973

basis

E

rHO

aVTZ aVQZ VDZF12 VTZF12 PES

0.0 0.0 0.0

0.973 0.971 0.971

OH + HBr 1.420 1.419 1.417

0.0

0.971

1.420

0.0

0.971

aVTZ aVQZ VDZF12 VTZF12 PES aVTZ aVQZ VDZF12 VTZF12 PES aVTZ aVQZ VDZF12 VTZF12 PES aVTZ aVQZ VDZF12 VTZF12 PES aVTZ aVQZ VDZF12 VTZF12 PES

rOH′

2.129

4.036

rH′Br

1.426

1.421

20.8

66.3

0.0

−0.33 −0.45 −0.34

3.954 1.418 Saddle (HOHBr‡) 0.975 1.504 1.479 0.972 1.507 1.478 0.972 1.507 1.477

101.9 102.1 101.9

141.3 140.7 139.6

55.4 54.3 55.4

−0.55

0.972

102.0

141.7

53.3

−0.52

0.973

1.516

1.478

105.7

143.7

73.7

−35.61 −36.01 −35.28

1.551 1.487 P-vdW (Br···H2O) 0.963 0.963 3.120 0.961 0.961 3.090 0.961 0.961 3.078

104.4 104.7 104.7

61.6 61.5 61.4

−103.9 −103.9 −103.9

−36.01

0.961

0.961

3.086

104.7

61.4

−103.9

−35.39

0.960

104.8

61.9

−103.5

−32.33 −32.63 −31.86

0.962 0.959 0.959

0.960 3.070 Br + H2O 0.962 0.959 0.959

−32.61

0.959

0.959

104.4

−32.36

0.965

0.965

107.3

104.2 104.4 104.3

the values of the spin-free PES and aVQZ values, −32.36 and −32.63 kcal mol−1, respectively. This ca. 10% difference is not expected to play a significant role in the dynamics calculations of the reaction cross section. Finally, we note that we examined the widely used T1 diagnostic and find that it is below 0.02 along most of the reaction path, but it does reach a maximum of 0.07 at the vicinity of the negative-energy saddle point. This latter value is somewhat larger than the “safe” 0.02 threshold; however, the UCCSD(T) calculations in that region of the PES do not produce obviously “spurious” results and so we feel reasonably confident that the final fitted PES is smooth, realistic and reliable. 708



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gelOH(T ) = 2(1 + e−|A| hc / kBT )

The QCT calculations were performed using standard procedures and expressions, 71,72 employing the code VENUS.73,74 The maximum impact parameter, (bmax), is strongly dependent on the collision energy for this reaction, ranging from 15.75 Å, in the limit of low collision energies, to 3.4 Å for the most energetic collisions. The vibration−rotation energy of the reactants was determined from the initial quantum states, using semiclassical quantization of the anharmonic diatom potentials from the PES. For a given set of initial parameters (collision energy, initial states of the reactants), the reactive cross section, using Monte Carlo sampling of the impact parameter, is given by 2 σr = πbmax

This product is indicated as “RR/SO” and if spin−orbit coupling is neglected, i.e., gOH el (T) is replaced by the factor 4, the result is indicated “RR/nSO”. A more accurate procedure to obtain the partition function involves a detailed quantum mechanical treatment of the energy levels in a fully coupled calculation.75,76 In this case, the partition function is given by, Ω= 3/2states OH Q coupled (T ) = 2[

+ e−|A| hc / kBT

(2)

(3)

where P(jOH; T) and P(jHBr; T) are the Boltzmann distributions for jOH and jHBr at temperature T; and then integrating over the thermal distribution of collision energies. The final expression for the thermal rate constant is given by,

∫

σr(Ec ; T )Ec 2

(kBT )

e−Ec / kBT dEc (4)

where kB is the Boltzmann constant, μ is the reduced mass for the collision (mOHmHBr/(mOH + mHBr)), ge is the electronic degeneracy of the reactive surface, and Qreact(T) is internal partition function of the reactants. The integral in eq 4 was calculated using a cubic spline interpolation of the averaged cross sections. The part of the reactant partition function describing the internal energies of the OH radical has received special attention in the literature, as its ground state is 2Π. Thus, there is both spin−orbit coupling and lambda-doubling to consider. Ignoring the latter, as is typically done in rate constant calculations, two versions of the rotational partition function have been used previously in rate constant calculations. The most common expression is given by a product of the spin-free rotational partition function times an electronic factor to account for the spin−orbit splitting of the Ω (the projection of the total electronic angular momentum on the molecular axis) = 1/2 and 3/2 states. Using rigid-rotor energy levels the factored partition function is given by Qrr(T) gOH el (T), where Q rr =

∑ (2J + 1)e−J(J+ 1)Bhc/k T B

J

(2J + 1)e−(EJ − EJ=1/2)/ kBT ] (7)

where EJ is the energy of a level with total angular momentum J, obtained from a numerical solution of the Schrödinger equation for the asymptotic fitted potential, calculated for states with Ω equal to 1/2 and 3/2, assuming that these states are separated by the experimental spin−orbit splitting constant,77 A. The details of these calculations and Hamiltonian are given in the SI. Results calculated using eq 7 for the partition function are indicated as “Coupled”. It could be argued that using the Coupled approach leads to an unbalanced treatment of rotational energies of OH in the partition function and as they are calculated by VENUS for the trajectories. In VENUS, as usual, they are obtained using a spinfree Hamiltonian, with the anharmonic diatom potential from the PES, and energies are obtained using standard and accurate EBK quantization. Dynamical consistency could be achieved by dynamically coupling the electronic and nuclear degrees of freedom either quantum mechanically, or with various schemes to couple nuclear and electronically degrees of freedom classically.78−80 However, that was not done in the present case, and so it might be argued even more strongly that since the present calculations are done using a single spin-free PES that any explicit consideration of spin−orbit coupling is inconsistent. In any case, the three expressions for the OH partition function will be used to calculate k(T). The thermal rate constants over the temperature range 5− 500 K obtained from QCT calculations and three different approaches for the partition function of the reactants are shown in Figure 3 along with available experimental data. The rate constant using reaction-path dual-level direct-dynamics variational transition-state theory21 is also shown. As seen, the present results with any treatment of the OH rotational partition function agree with experiment in showing a nearly constant k(T) at temperatures above roughly 180 K and a strong inverse temperature dependence below roughly 150 K. As also seen, the choice of OH-partition function does begin to play a major role below 150 K, where factors of between 2 and 4 are seen. As expected, the largest differences are at the lowest temperatures. Between 50 and 150 K, the RR/SO, RR/nSO, and Coupled results overestimate experiment on average by factors of 2.2, 1.2, and 1.5, respectively. Below 50 K, the RR/SO rate constants overestimate the measurements by a factor of 1.5, and the RR/nSO and Coupled calculations underestimate experiment by factors of 1.3 and 1.6, respectively. Overall, the best agreement with experiment is for the RR/nSO results, where the differences are roughly 20%. This level of agreement is probably fortuitously good.

σr(Ec , jOH , jHBr )P(jOH ; T )P(jHBr ; T )

ge 8kBT μπ Q react(T )

∑ J = 1/2

jOH , jHBr

k(T ) =

(2J + 1)e−(EJ − EJ=3/2)/ kBT

J = 3/2

where Nr is the number of reactive trajectories and Ntotal is the total number of trajectories. The initial state-specific cross sections were calculated for 30 values of collision energies (Ec) ranging from 0.001 to 25 kcal mol−1, with the semiclassical rotational quantum numbers ranging from 0 to 7, for both reactants in the vibrational ground state. The very low collision energy was needed to converge the rate constant to temperaturea as low as 5 K. The rate constant can be calculated by averaging the reactive cross section with respect to the population of the initial states

∑

∑

Ω= 1/2states

Nr Ntotal

σr(Ec ; T ) =

(6)

(5)

and 709



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in the QCT rate constant, however, is not evident in the experimental results, measured down to 23 K, and clearly it would be important for experiment to go below that temperature. It may be that the self-reaction OH + OH81,82 may affect to some extent the measurement of the OH + HBr rate constants at very low temperatures. To summarize, a new ab initio potential energy surface has been reported and used in quasiclassical calculations of the rate constant of the OH + HBr → Br + H2O reaction, which shows an unusual temperature dependence. The theoretical results are (finally) in good agreement with experiment. Further investigation of the product state distribution, isotope effects on the rate constants and in the energy disposal, and transitionstate theory calculation are being conducted and will be presented in a forthcoming paper. Finally, with the availability of the global PES, it would be interesting, albeit challenging, to conduct quantum calculations of the rate constant to investigate the importance of tunneling, e.g., through rotationally adiabatic barriers.

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ASSOCIATED CONTENT

S Supporting Information *

Benchmark calculations of the energetics, harmonic vibrational frequencies of the stationary points, calculated and experimental thermal rate constants, and details of the determination of the energies used in the coupled partition function calculation. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Figure 3. Comparison of the thermal rate constants calculated for the OH + HBr reaction with the available experimental data, indicated as symbols (refs 3−14), and with the reaction-path dual-level direct dynamics variational transition-state theory calculations from Liu et al.21

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS F.R.O. acknowledges the Conselho Nacional de Desenvolví e Tecnológico (CNPq) of Brazil for academic mento Cientifico support. A.G.S.O.F. is thankful to Fundaçaõ de Amparo à Pesquisa do Estado de São Paulo (FAPESP). J.M.B. thanks the Department of Energy (DE-FG02-97ER14782) for financial support.

Note also that the three approaches give a maximum that is located at roughly 15 K for the RR/SO and RR/nSO calculations, and 50 K for the Coupled calculation. The existence of the maximum is in agreement with a prediction made by Clary et al.17 for reactions of molecules in 2Π electronic states with 1Σ polar molecules at low temperatures. Clary et al. did consider the OH + HBr reaction in the range 0 to 80 K, using adiabatic capture theory.17 The reported rate constant is in the range 30−35 × 10−11 cm3 molecule−1 s−1, which is significantly larger than the present calculated results and experiment. This is perhaps due to the assumption in the capture model of a unit reaction probability for collisions with impact parameters less than the predicted maximum one. At low collision energies, the QCT results find a smaller reaction probability owing to the formation of quasi-trapped trajectories due to the prereactive van der Waals well. A significant fraction of such trajectories emerge to reform the reactants. (Details of these trajectories will be given elsewhere.) Next consider the previous reaction-path dual-level directdynamics variational transition state theory calculations of Liu et al.21 As seen, they significantly underestimate experiment at temperatures above 60 K. They do increase very rapidly below that temperature, and no evidence of a temperature maximum is seen. Overall, the present results represent a major improvement of the theoretical modeling of the OH + HBr reaction. The level of agreement with experiment is very good. The maximum

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REFERENCES

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Quasiclassical Trajectory Calculations of the Rate Constant of the OH

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