New Stress Test for Ring Polymer Molecular Dynamics: Rate

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Article

New Stress Test for Ring Polymer Molecular Dynamics: Rate Coefficients of the O(P)+HCl Reaction and Comparison with Quantum Mechanical and Quasiclassical Trajectory Results 3

Marta Menendez, Pablo G. Jambrina, Alexandre Zanchet, Enrique Verdasco, Yury V. Suleimanov, and F. Javier Aoiz J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b06695 • Publication Date (Web): 28 Aug 2019 Downloaded from pubs.acs.org on August 30, 2019

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The Journal of Physical Chemistry

New stress test for Ring Polymer Molecular Dynamics: Rate Coecients of the O( P )+HCl Reaction and Comparison with Quantum Mechanical and Quasiclassical Trajectory Results 3

∗,†

M. Menéndez,

P. G. Jambrina,

Suleimanov,

‡

∗,¶,§

†

A. Zanchet,

E. Verdasco,

and F. J. Aoiz

†

Y. V.

†

†Departamento de Química Física I, Facultad de Ciencias Químicas, Universidad

Complutense de Madrid , 28040 Madrid, Spain ‡Departamento de Química Física, Facultad de Ciencias Químicas, Universidad de

Salamanca, 37008 Salamanca, Spain ¶Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kava

Strasse, Nicosia 2121, Cyprus §Department of Chemical Engineering, Massachusetts Institute of Technology, 77

Massachusetts Avenue, Cambridge, Massachusetts 02139, United States E-mail: [email protected]; [email protected]

Abstract In the last decade, Ring Polymer Molecular Dynamics (RPMD) has emerged as a very ecient method to determine thermal rate coecients for a great variety of

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 35

chemical reactions. This work presents the application of this methodology to study the O(3 P ) + HCl reaction, which constitutes a stringent test for any dynamical calculation due to rich resonant structure and other dynamical features. The rate coecients, calculated on the 3 A0 and 3 A00 potential energy surfaces (PESs) by Ramachandran and Peterson [J. Chem. Phys. 2003, 119, 9590], using RPMD and quasiclassical trajectories (QCT) are compared with the existing experimental, and the quantum mechanical (QM) results by Xie

et al.

[J. Chem. Phys. 2005

, 014301]. The agreement is

122

very good at T > 600 K, although RPMD underestimates rate coecients by a factor between 4 to 2 in the 200500 K interval. The origin of these discrepancies lies on the large contribution from tunneling on the 3 A00 PES, which is enhanced by resonances due to quasibound states in the van der Waals wells. Although tunneling is fairly well accounted for by RPMD even below the crossover temperature, the eect of resonances, a long-time eect, is not included in the methodology. At the highest temperatures studied in this work, 2000-3300 K, the RPMD rate coecients are somewhat larger than the QM ones, but this is shown to be due to limitations in the QM calculations and the RPMD are believed to be more reliable.

Introduction Craig and Manolopoulos proposed an elegant and simple way to incorporate quantum mechanical eects of nuclear motions in real-time dynamics by using classical molecular dynamics in an extended phase space of ring polymer consisting of classical copies of the original system, which interact with neighboring beads in the necklace via ctions temperaturedependent harmonic potentials.

1

The idea of classical (ctions) dynamics of ring polymer

was originally proposed a few decades ago and was used to calculate various static thermodynamic properties as one can show (via so-called classical isomorphism)

2

that classical

partition function of this ring polymer, as well as other thermodynamic averages, rigorously converges to quantum mechanical partition function. Craig and Manolopoulos proposed to

2


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use the classical isomorphism to approximate real-time correlation functions, which are used to describe many dynamical properties of systems in thermal equilibrium (such as diusion coecients, infrared absorption spectra, rate coecients). Though introduced in a rather

ad hoc

fashion and lacking its rigorous derivation,

35

this approach, named ring polymer

molecular dynamics (RPMD), gives the correct quantum mechanical result in various limiting cases (high-temperature and short-time limits, interaction via harmonic oscillator, static equilibrium).

1

RPMD preserves the quantum Boltzmann distribution and time-reversal sym-

metry and also scales favorably with dimensionality of the system as it is based on purely classical molecular dynamics.

3

For thermal rate theory, RPMD provides additional desirable

properties of being exact for tunneling through a parabolic barrier of the transition state dividing surface

7

6

and of being independent

that is used in the rate theory formalism to separate

reactant(s) from product(s). Manolopoulos and colleagues applied RPMD to various systems with the main focus on condensed phase and found that it provides ecient and reasonably accurate way to study quantum mechanical eects in real-time dynamics of nuclear motions as described in their review.

3

Later on, Suleimanov and colleagues found that RPMD exhibits excellent

performance when used to calculate thermal chemical reaction rate coecients. They have analyzed various prototype gas-phase chemical reactions including (a) thermally activated atom-diatom chemical reactions, from the simplest possible H + H2 reaction isotopologues

9

8

and all its

to stress test systems such as D + MuH (for quantum tunneling),

H2 (for the zero-point energy eect),

11

F + H2 (for high reaction asymmetry),

(for heavy-light-heavy mass combination); reactions (X = H

12

and its isotopes,

13,14

8

O,

(b) polyatomic X + CH4 /CD4 /

15,16

Cl,

17

and OH;

18,19

13

8

10

Mu +

Cl + HCl

CH4 chemical

(c) atom-diatom insertion

1 20 2 20 1 21,22 1 21 reactions X + H2 , where X stands for O( D), N( D), C( D), S( D); (d) polyatomic chemical reactions with complex reaction paths (with both barrier and minimum).

23,24

These

studies were mainly focused on method assessment as most of them used analytical potential energy surfaces (PESs) so as to enable comparison with the rigorous quantum mechanical

3



Page 4 of 35

(QM) results and other dynamics approximations available for those PESs. The results of this method assessment have been summarized in the recent review by Suleimanov, Aoiz and Guo.

25

In brief, for thermally activated reactions, it has been con-

cluded that RPMD provides very reliable and consistent performance: (a) RPMD is exact in the high-temperature limit; (b) it is reliable at intermediate temperatures (with deviations from QM not far from the convergence errors); (c) at low temperatures, below the so-called crossover temperatures in the deep-tunneling regime, RPMD systematically overestimates (underestimates) the rate coecients by a factor of 2-3 for energetically asymmetric (symmetric) chemical reactions.

8,9,26

Such systematic behavior of RPMD rate theory con-

trasts with other approximations based on transition state theory (TST) or quasiclassical trajectories (QCT), as becomes evident when applying them to various stress systems, especially at low temperatures where quantum mechanical eects play a dominant role.

In

particular, RPMD demonstrated its robustness for one of the most challenging atom-diatom chemical reactions of Mu-transfer in D + MuH, for which two most representative TSTbased approximations, semiclassical instanton and canonical variational TST (CVT), failed to provide accurate results at low temperatures in the deep tunneling regime.

10

One of the

main advantages of RPMD over TST is the independence of the former from the choice of the transition-state dividing surface,

7

proper identication of which becomes increasingly

dicult as the dimensionality of the problem rises due to the multidimensional nature of tunneling at low temperatures, even for such as simple atom-diatom chemical reaction as D + MuH.

10

Another reason why RPMD is accurate in the deep-tunneling regime stems

from the fact that it gives the exact

t=0

surface between reactants and products, instanton.

27

quantum ux through a path-integral dividing

which correctly describes uctuations around the

26

Comparison with the QM results and experiment for prototypical thermally activated polyatomic chemical reactions (the most representative references are 12,15,16,18) conrmed that the sensitivity of TST-based methods to the choice of the transition-state dividing sur-

4


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face leads to less accurate results in higher dimensionalities and, moreover, it was found that more elaborate quantum implementations of TST do not necessarily provide improved accuracy when compared to less sophisticated TST methods. It was also surprising to observe that TST-based methods may experience issues at high temperatures due to anharmonicity eects near the saddle point region, which is often treated harmonically.

18

Because RPMD

preserves the quantum Boltzmann distribution (and as result the ZPE) along the reaction path, it is immune to this issue. For insertion reactions, RPMD demonstrated even more remarkable accuracy being within no more than

15% from the rigorous quantum dynam-

ics results for all four prototype atom-diatom chemical reactions.

20,21

Overall, application of

RPMD to prototypical gas phase chemical reactions demonstrated its unique nature of being reliable and accurate across multifarious scenarios (reaction coordinate prole, mass combinations, temperatures etc.). Due to the success of the RPMD rate theory, recent RPMD studies are shifting to polyatomic and possibly more complex chemical reactions.

2831

It is known that the RPMD methodology is reliable at short-times (up to the thermal time

∼ β¯ h) 32

but it does not include long-time quantum eects such as interference. While

some previously studied reactions do exhibit single resonances at very low collision energies or even more pronounced resonant structure that could potentially aect thermal rate coecients of typical temperature range of interest in the RPMD studies (above half of the crossover temperature),

25

the previous method assessment misses an explicit analysis

of chemical reaction with strong resonant structure of the reaction probabilities due to the reaction intermediate.

3 The O( P ) + HCl reaction appears as an interesting benchmark for the RPMD method. This reaction has been subject of a series of dynamical studies

3337

since the advent of accurate PESs by Ramachandran and coworkers. take place adiabatically on two PESs of symmetry ground state, OH( the

3

A0

2

Π) + Cl(2 P ), products.

is collinearly dominated, the

3

A00

over the last 15 years

38,39

The reaction may

A0 and A00 that correlate with reagents and

However, the two PESs are vastly dierent: while

PES, with a signicantly lower barrier, has a bent

5



Page 6 of 35

transition state. In eect, most of the reactivity at temperatures below 1000 K proceeds from the

3

A00

PES with almost a negligible contribution from the

3

A0

PES, although at higher

temperatures it may amount up to 20% of the total reaction yield.

In addition, the

3

A00

PES evinces the presence of fairly deep van der Waals (vdW) wells that result in a very dense resonance structure which aects reaction probabilities, as it will be shown below. The present RPMD calculations are aimed to assess the accuracy of the method on these two very dierent PESs on which tunneling plays a major role.

QM and RPMD results

will also be compared with QCT results obtained using the cumulative reaction probability formulation.

40,41

The paper is laid out as follows. The details of RPMD, QCT and QM calculations carried out in this work will be presented rst. The results, will be presented in the Section Results, and will be discussed in the Discussion Section. Finally, the conclusions of this work will be presented in the closing section.

Computational details The present dynamical study on the O(

3

P)

+ HCl(X

1

Σ) →

been carried out separately on the two adiabatic PESs,

3

2 2 OH( Π) + Cl( P ) reaction has

A00

and

3

A0 ,

that correlate with

the same reagents and products. We have used the PESs calculated by Ramachandran and Peterson.

39

However, it was found that the QCT and RPMD calculations on the original

t were too slow, since they require the evaluation of the gradients in each integration step (the original t does not include analytical derivatives).

For this reason, we generated a

new t using the Reproducing Kernel Hilbert Space (RKHS) ansatz, following the procedure described in ref 42. Specically, the analytical PESs by Ramachandran and Peterson were used to compute the energies over a new set of geometries sampled over a regular grid for the two PESs.

43

The many-body decomposition was then used to separate the contributions

of the two and three body terms and then tted independently. The two-body terms were

6


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obtained by calculating the three involved diatomics molecule considering the third atom to a distance larger than 100 Å. The resulting PESs was carefully checked by comparing the potential at a long series of conguration points and, in addition, QM and QCT calculations were performed on both PESs leading to essentially indistinguishable results. RPMD calculations have been performed using the strategy is documented in the

RPMDrate

rate theory and its practical applications.

manual

44

RPMDrate code. 44 The computational

and in the recent review of the RPMD

25

The simulation parameters are similar to those used in the previous RPMD calculations for thermally activated chemical reactions.

11,23

For the

3

A0

surface, calculations were carried

out using 64 ring polymer beads, although 32 beads were enough to achieve convergence for the lowest temperature studied of 200 K. For the

3

A00

PES, calculations were performed

using 128 beads, although, except for the lowest temperature, the results with only 32 beads were essentially the same. Purely classical calculations (not QCT) correspond to setting the number of beads,

nb ,

to one.

45

For the calculation of the potential of mean force (PMF) proles, in this occasion the reaction coordinate

0.02

width.

45

ξ ∈ [−0.05, 1.05],

dened elsewhere,

8,11,25

was divided in windows of

In each window 200 constrained RPMD trajectories were run with 20 ps of

thermalization time and 100 ps of running time, using a time step of 0.1 fs. The constraint that we apply at the window centered at type

k(ξ − ξi )2

consists in adding a harmonic potential of the

to the Hamilton function of the system, so that the trajectory explores

only the vicinity of

ξi .

The value of the force constant

explore the surroundings of between neighboring window,

ξi

ξi

k

must be chosen big enough to

and at the same time small enough to allow overlapping

ξ distributions.

With the rst two moments of the

ξ distribution at each

hξi i and hξi2 i, we can apply a thermodynamic integration scheme 46,47 to compute the

centroiddensity quantum transition state theory (QTST) rate coecient. The RPMD rate coecients are obtained combining the QTST rates with the time-dependent ring polymer transmission coecients (or ring polymer recrossing factor). For the recrossing calculations,

7



a total of 10

5

Page 8 of 35

child trajectories were run starting from an initial parent trajectory of 20 ps.

Each child trajectory was run for 0.1 ps. To compare with the RPMD rate coecients and to provide further insight into the dynamics of the reactions, QCT and QM calculations were also carried out.

Instead of

using the standard QCT calculations selecting randomly collision energies and internal states according to Maxwell-Boltzmann distributions at various temperatures, we have computed QCT following the cumulative reaction probability formalism described in refs 40 and 41. Two kinds of calculations were carried out.

In the rst, cumulative reaction probabilities

(CRPs) were calculated as a function of total energy, angular momentum values,

J , CrJ (Etot ).

Etot ,

for specic values of the total

The internal states of the HCl (and hence the

collision energy) were microcanonically sampled. In this manner, QCT and QM CRPs can be directly compared. Batches of

2.5 × 106

trajectories for selected

eV total energy range were run on the two PESs.

J

values in the 0.41.7

The second kind of calculations were

aimed to determine the thermal cumulative reaction probabilities and rate coecients. In this case, the total energy was selected uniformly in the 0.44.0 eV range of total energies to cover Boltzmann distributions at 3500 K, the highest temperature here studied. Once the total energy had been selected, the remaining collision parameters (HCl rovibrational state, total angular momentum and its projection onto the internuclear axis, impact parameter and molecular orientation) are uniformly sampled. found in ref 40. For these calculations, a total of 10

The details of the calculations can be

7

trajectories were run in the 0.44.0 eV

range of total energies. Each individual trajectory was integrated between initial and nal points at which the distance between the atom and the center of mass of the diatomic was 8 Å; the integration time step (0.05 fs) was such that energy conservation was better than 1 in

105 .

The initial rovibrational energies were calculated semiclassically using the asymptotic

diatomic potential energy of the PESs; they agree with their exact, quantum counterparts to within four signicant gures. QM calculations of CRPs were carried out on both PESs using the coupled-channel

8


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hyperspherical coordinates formalism with the ABC code.

48

Calculations were performed

for the two triatomic parities and total angular momentum values

J = 0, 20, 50, 80

and

150,

and total energies in the 0.2-1.7 eV range. The propagation was performed in 120 sectors with a maximum hyperradius levels up to Xie

max Etot =2.5

et al., 36

then they used the standard 200 with

eV,

ρmax = 16 a0 ,

jmax =99,

including in the basis all the diatomic energy

and helicity quantum numbers up to

carried out similar calculations on the

J -shifting

for

J > 10.

3

A0 3

Calculations on the

for

A00

J ≤

Kmax = 12.

10 and

Kmax =10

and

were mainly performed using

J -shifting approximation but CRPs were also calculated for J =20, 70, 130 and

max =2.75 Kmax =15, Etot

eV, and a maximum rotational quantum number

jmax =25.

For comparison purposes, in the present work two additional sets of calculations were run only on the

3

A00

PES for

J =0

at 250 total energies in the 0.2-2.7 eV range, with maximum

internal energy up to 3.5 eV. In the rst batch the maximum rotational quantum number of any channel was limited to

jmax =25,

as in ref 36, while in the second one,

will be shown, the dierences are remarkable at total energies

Etot >1.8

jmax = 99.

As it

eV.

Once the QM and QCT CRPs have been obtained, the thermal rate coecient,

k(T ),

can be derived using the following expression

Z

∞

Cr (Etot )exp(−Etot /kB T ) dEtot k(T ) =

0

(1)

hΦrel (T ) QHCl v,j (T )

where the total CRP as a function of the total energy,

Cr (Etot ) =

JX max

Cr (Etot ),

is given by

(2J + 1) CrJ (Etot )

(2)

J=0

with

kB

CrJ (Etot ) indicating the CRP for a given value of the total angular momentum.

is the Boltzmann constant, and

Φrel (T )

and

QHCl v,j (T )

are the relative translational and

HCl rovibrational partition functions, respectively.

9

In eq. 1,



The thermal-CRP,

T -CRP,

Page 10 of 35

including the reagent's partition function, is dened as

Cr (Etot , T ) =

Cr (Etot )exp(−Etot /kB T ) hΦrel (T )QHCl v,j (T )

whose integration over the total energy range yields

k(T ).

The

(3)

T -CRP represents the contri-

bution of the Boltzmann weighted total CRP to the rate coecient in an interval of energies within

Etot

and

Etot + dEtot . 40

To account for the spin-orbit splitting of the oxygen atom and saddle point electronic state, the rate coecients have been multiplied by a factor to represent the fraction of collisions that correlate with the reactive

f (T ) =

At high temperatures, the

3

A0

or

3

A00

f (T )

3

A0

or

3

A00

PESs, given by

3 5 + 3 exp(−227.7/T ) + exp(−326.6/T )

(4)

is roughly 1/3, which represents the ratio of the degeneracy of

surface (3) and

3

P

oxygen atom (9).

36

Results Figure 1 shows the prole of the minimum energy path (MEP), as well as the vibrationally adiabatic curves (including the zero point energy) for both PESs as a function of the mass scaled reaction coordinate,

s.

The coordinate

s

is a measure of the progress of the reac-

tion and is calculated as the distance from the saddle point. In terms of the mass scaled coordinates

Q1

and

Q2 ,

the modulus of

s

is dened as:

49,50

h i1/2 |s| = (Q1 − Q1 SP )2 + (Q2 − Q2 SP )2

10


(5)

Page 11 of 35

which are related to the internuclear distances for an A + BC

→

AB + C reaction through

h i mC 1/2 Q1 = µA−BC RAB + RBC mBC

(6)

1/2

Q2 = µBC RBC ,

µBC

and

tively;

µA−BC

(7)

are the reduced masses of the reactant diatom and atom-triatom, respec-

mC and mBC are the masses of the C atom and BC diatom, respectively. RAB and RBC

are the AB (OH) and BC (HCl) internuclear distances. In eq 5,

Q1 SP and Q2 SP are the values

of the mass scaled coordinates that dene the saddle point. The sign of

s

is chosen arbi-

0 .8

3

A "

V (e V )

0 .6

M E P + Z P E

0 .4 0 .2 0 .0

M E P

-0 .2 -4 .0

-2 .0

0 .0

2 .0

4 .0

0 .8

3 0 .6

M E P + Z P E

A '

V (e V )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .4

0 .2

M E P 0 .0 -4 .0

-2 .0

0 .0

s ( u

1 /2

2 .0

4 .0

Å )

Figure 1: Minimum energy paths (dashed lines) as a function of the mass-scaled reaction 3 3 0 coordinate (see text for denition) for the O( P )+HCl reaction on the A (bottom panel) 3 00 and A (upper panel). The reaction paths including their respective zero point energies are also shown in the gure as solid lines. The van der Waals wells in the entrance and exit 3 00 channels of the A are evident in the gure.

11



Page 12 of 35

trarily, and positive/negative values of s are used for the description of the product/reactant valleys.

50

As commented on above, the TS of that of the

3

A00

3

A0

PES corresponds to a collinear approach, while

◦ PES is strongly bent (135 ). Apart from the remarkable dierence in their

barrier heights, 0.597 eV (0.778 eV with the ZPE) on the the ZPE) on the

3

A00

3

A0 and 0.459 eV (0.564 eV including

measured from the minimum of the O(

most salient feature is the presence on the

3

A00

3

P )+HCl

asymptote, the next

PES of fairly deep vdW wells in both the

entrance and exit channels of 0.07 eV and 0.24 eV depths, respectively, with respect to

3 the O( P )+HCl asymptote. Moreover, the minimum in the exit vdW well is strongly bent ◦ 39 (66 ). Such wells may cause the formation of short-lived intermediate collision complexes leading to quantum scattering resonances that aect the rate coecients, as it will be shown below. We will rst discuss the calculation carried out using the RPMD method. Figure 2 shows the potential mean force proles, on the two PESs. smaller on the coecients,

3

A00

W (ξ), 8,11,25

calculated at 200 K, 500 K, 1000 K and 3000 K

As can be seen, the maximum value of

W (ξ)

at each temperature is

PES revealing a lower reaction barrier. The time-dependent transmission

κ(t), are shown in the lower panels of Figure 2.

They are similar for

and exhibit the same temperature dependence. It is interesting how smaller than at 500 K and 1000 K at

t >10

κ(t)

3

A0

and

3

A00

at 3000 K becomes

fs, indicating an enhancement of recrossing with

increasing temperature. A summary of the results from RPMD calculations for the newly tted

3

A0

and

3

A00

PESs

for several ring polymer beads, (nb ), can be found in Tables 1 and 2, respectively, including the QTST rate coecients and the recrossing factors, displays the RMPD

3

A0

k(T )

κ(t),

for

t ≥ 40f s.

The last column

including the electronic partition function factor, eq 4.

For the

PES, it can be seen in Table 1 that 32 beads are enough to achieve convergence for the

lowest temperature studied of 200 K, although calculations have been made using 64 beads. For the

3

A00

PES, also 32 beads seems to ensure convergence to get an accuracy of around

12


Page 13 of 35

2 .0

2 .0

3

W ( ξ) / e V

1 .5

3

A '

1 .5

1 .0

1 .0

0 .5

0 .5

0 .0

3 0 0 1 0 0 5 0 0 2 0 0

A "

0 K 0 K K K

0 .0 0 .0

0 .2

0 .4

ξ

0 .6

0 .8

1 .0

0 .2

0 .0

0 .4

0 .6

0 .8

1 .0

ξ

1 .0

1 .0

0 .8

0 .8

0 .6

0 .6

κ( t )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3 0 .4 0

3

A ' 1 0

2 0

3 0

4 0

0 .4

A "

0

1 0

2 0

3 0

4 0

t (fs )

t (fs )

Figure 2: RPMD potential of mean force proles (top panels) and time dependent trans3 3 0 mission coecients (bottom panels) for the O( P )+HCl reaction on the A (nb =64) (left 3 00 panels) and A (nb =128) (right panels) PESs at 200 K (black solid), 500 K (blue dash), 1000 K (red dashdot), and 3000 K (green dashdot-dash).

10 %, although these calculations have been upgraded using

nb = 128

(256 at 200 K). As

it could be expected, at the highest temperature, 3000 K, the purely classical calculations,

nb = 1,

on both PESs are not far from those obtained with a converged number of beads.

The temperature dependent RPMD rate coecients on

3

A0

and

3

A00

PESs, whose values

are listed in the rightmost columns of tables 1 and 2, are shown as Arrhenius plots in the left and right panels, respectively, of Figure 3. At 200 K the value of

k(T )

on the

3

A00

PES

3 3 0 is almost a factor of 2·10 larger than on the A PES. Only at the highest temperatures,

T >1000 K

the contribution of the latter becomes signicant.

For comparison, the rate

coecients calculated by QCT in the present work and those from the QM calculations of

13



Table 1:

Page 14 of 35

3 3 0 RPMD Rate coecients for the reaction O( P) + HCl, computed in the A PES.

The parenthesis denote powers of ten.

The last column numbers include the factor that 3 0 A PES (eq 4). Units are in

accounts for the fraction of collisions that correlate with the 3 −1 cm s .

T /K

nb

k QTST

κ(t)

k RPMD

f (T ) k RPMD

200

1

3.443(-26)

0.471

1.622(-26)

200

8

5.424(-21)

0.104

5.641(-22)

200

16

1.773(-21)

0.535

9.486(-22)

200

32

1.673(-21)

0.607

1.016(-21)

200

64

1.657(-21)

0.635

1.052(-21)

5.127(-22)

200

128

1.670(-21)

0.634

1.059(-21)

5.160(-22)

250

1

1.321(-23)

0.517

6.830(-24)

250

64

3.510(-20)

0.687

2.411(-20)

300

1

1.295(-21)

0.550

7.123(-22)

300

64

4.384(-19)

0.720

3.156(-19)

400

1

4.090(-19)

0.645

2.638(-19)

400

64

1.888(-17)

0.750

1.416(-17)

500

1

1.725(-17)

0.656

1.132(-17)

500

64

2.480(-16)

0.756

1.875(-16)

600

1

2.195(-16)

0.678

1.488(-16)

600

64

1.600(-15)

0.761

1.218(-15)

800

1

5.744(-15)

0.700

4.021(-15)

800

64

1.924(-14)

0.742

1.428(-14)

1000

1

4.352(-14)

0.704

3.064(-14)

1000

64

9.784(-14)

0.726

7.103(-14)

2000

1

3.264(-12)

0.687

2.242(-12)

2000

64

4.034(-12)

0.703

2.836(-12)

3000

1

1.684(-11)

0.659

1.110(-11)

3000

64

1.677(-11)

0.692

1.160(-11)

1.117(-20) 1.405(-19) 5.950(-18) 7.577(-17) 4.786(-16) 5.406(-15) 2.627(-14) 9.978(-13) 4.012(-12)

ref 36 are also displayed in Figure 3. The QM data on both PESs are systematically bigger than the RPMD results on both PES except at the highest temperatures, where the QM

k(T )

lie below the RMPD ones, an eect that will be discussed below.

results, only above

T >600 K

As for the QCT

their values become comparable to those given by RPMD, as

tunneling, which is at least in part captured by RPMD, plays an important role. highest temperatures calculated here (T

>1000

K) the values of the QCT and RMPD

are similar, but the convergence is somewhat slower on the

14

At the


3

A00

PES.

k(T )

Page 15 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2:

3 3 00 RPMD Rate coecients for the reaction O( P) + HCl, computed in the A PES.

The parenthesis denote powers of ten.

The last column numbers include the factor that 3 00 A PES (eq 4). Units are in

accounts for the fraction of collisions that correlate with the 3 −1 cm s .

T /K

nb

k QTST

κ(t)

k RPMD

200

1

6.856(-22)

0.801

5.492(-22)

200

8

3.771(-18)

0.180

6.788(-19)

200

16

2.656(-18)

0.540

1.434(-18)

200

32

2.964(-18)

0.590

1.749(-18)

200

64

3.074(-18)

0.600

1.844(-18)

200

128

3.175(-18)

0.603

1.915(-18)

9.330(-19)

200

256

3.142(-18)

0.602

1.891(-18)

9.217(-19)

250

1

4.902(-20)

0.802

3.931(-20)

250

128

3.080(-17)

0.660

2.033(-17)

300

1

1.232(-18)

0.800

9.856(-19)

300

128

1.933(-16)

0.694

1.342(-16)

400

1

1.010(-16)

0.790

7.979(-17)

400

128

2.788(-15)

0.725

2.021(-15)

500

1

1.436(-15)

0.767

1.101(-15)

500

128

1.675(-14)

0.725

1.214(-14)

600

1

9.891(-15)

0.760

7.517(-15)

600

128

6.198(-14)

0.745

4.618(-14)

800

1

1.219(-13)

0.721

8.789(-14)

800

128

3.595(-13)

0.756

2.718(-13)

1000

1

5.640(-13)

0.693

3.909(-13)

1000

128

1.192(-12)

0.760

9.059(-13)

2000

1

1.484(-11)

0.608

9.023(-12)

2000

128

1.815(-11)

0.718

1.303(-11)

3000

1

5.122(-11)

0.567

2.904(-11)

3000

128

5.644(-11)

0.673

3.798(-11)

f (T ) k RPMD

9.415(-18) 5.970(-17) 8.493(-16) 4.908(-15) 1.815(-14) 1.029(-13) 3.351(-13) 4.585(-12) 1.313(-11)

Table 3 contains the total RPMD and QCT thermal rate coecients (summing the contributions of both PESs) as a function of the temperature. These results are compared with the QM and the semiclassical treatment of improved canonical variational TST including tunneling on the rate constant (ICVT/µOMT) data from ref 36. The column with corresponds to the empirical expression:

k(T )exp

56

k(T ) = (9.27 ± 0.03) × 10−24 × T 3.67±0.18 exp[−(1030 ± 160)/T ] 15


(8)


1 0

-1 0

1 0

-1 2

1 0

-1 4

1 0

-1 6

1 0

-1 8

1 0

-2 0

1 0

-2 2

1 0

-2 4

s -1 ) 3

k ( T ) (c m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3

A '

Q C T R P M D Q M 1

1 0 0 0 /T (K 2

3

4 -1

5

1 0

-1 0

1 0

-1 2

1 0

-1 4

1 0

-1 6

1 0

-1 8

1 0

-2 0

3

Page 16 of 35

Q C T R P M D Q M

A "

1 0 0 0 /T (K 2

1

)

3

4 -1

5

)

3 0 Figure 3: Comparison of RMPD, QCT and QM (from ref. 36) rate coecients on A (left) 3 00 and A (right) PESs. The factor that represents the fraction of collisions that correlates 3 0 3 00 with the reactive A and A , eq 4, is included in all cases. Since the QM k(T ) data on 3 0 the A PES are not explicitly given in ref. 36, they have been calculated as the dierence 3 00 between the tabulated sum of the rate coecients on both PES and those on the A PES. 3 0 Given the relatively small contribution of the A to the overall reaction, only values at

T ≥500 K

can be reliably obtained from the dierence.

3 3 0 3 00 Table 3: Comparison of the O( P )+HCl total (summing the contributions from A and A PESs) rate coecients calculated using the RPMD, QCT, QM and ICVT/µOMT methods. The QM and ICVT/µOMT data are taken from ref 36. All 3 −1 The parenthesis denote powers of ten. Units are in cm s .

k(T )

contain the factor,

T (K)

k(T )RPMD

k(T )QCT

k(T )QM

k(T )ICVT/µOMT

k(T )exp

200

9.223(-19)

1.714(-20)

3.72(-18)

1.68(-19)

1.50(-17)

300

5.984(-17)

1.138(-17)

1.84(-16)

2.88(-17)

3.69(-16)

400

8.553(-16)

3.300(-16)

2.01(-15)

5.79(-16)

2.50(-15)

500

4.984(-15)

2.689(-15)

1.00(-14)

4.10(-15)

9.50(-15)

600

1.863(-14)

1.151(-14)

3.22(-14)

1.64(-14)

2.61(-14)

800

1.083(-13)

7.868(-14)

1.59(-13)

1.06(-13)

1.15(-13)

1000

3.614(-13)

2.715(-13)

4.62(-13)

3.54(-13)

3.39(-13)

2000

5.583(-12)

4.639(-12)

5.54(-12)

4.26(-12)

7.21(-12)

3000

1.714(-11)

1.481(-11)

1.44(-11)

9.62(-12)

3.79(-11)

16


f (T ).

Page 17 of 35

1 0

-1 3

1 0

-1 5

H a c k e t a M a h m u d S in g le to n B ro w n & S C .L in e t a

l. e t a l. e t a l. m ith l.

Q M Q C T I C V T / µO M T R P M D

1 0

-1 7

1 0

-1 9

1 0

-1 0

1 0

-1 1

1 0

-1 2

1 .0

2 .0

3 .0

4 .0

5 .0

s

-1

)

k (T ) (c m

3

s

-1

)

1 0

-1 1

3

k (T ) (c m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .3

0 .4

0 .5

0 .6

0 .7

1 0 0 0 /T (K Figure 4:

0 .8 -1

0 .9

1 .0

)

Comparison of experimental rate coecients with those obtained with RPMD

(black line with solid circles), QCT (red solid line) and QM (green short-dash line) from ref 3 36 for the O( P )+HCl reaction between 200 and 3200 K. Symbols represent the experimental results from refs 5156. The bottom panel is a blowup of the inset of the upper panel covering the 1000-3200 K temperature range

The results from table 3 are also shown as an Arrhenius-type plot in Figure 4, along with the available experimental data.

5156

The lower panel of the gure is an enlargement of the

high temperature region (1000 K3200 K) of the results. In general terms, there is a fair agreement between the RPMD and the QM calculations from ref 36.

The crossover temperature,

Tc = h ¯ νa /kB ,

17

−1 is 421 K, where iνa =1840.7i cm



is the imaginary frequency at the barrier maximum of the surprising that at

T =200 K

a factor of four.

Even at 500 K, slightly above

Page 18 of 35

3

A00

PES.

39

Therefore, it is not

the RMPD calculation underestimates the rate coecient by

Tc ,

there is a dierence of a factor of 2.

Considering that the RPMD method has demonstrated a consistently good performance for symmetric thermally activated reactions in the deep tunneling regime, it would not be expected discrepancies larger than a factor of 2 even in the deep tunneling regime. In contrast, at the highest temperatures,

T >2000 K the RMPD k(T ) is almost 20% bigger than

the QM values from ref 36. It must be pointed out that the the high-temperature QM results presented in Table 3 and Figure 4 have been obtained by interpolating the corresponding cumulative reaction probabilities. This may be a rather crude approximation, and therefore QM deviates from RPMD, which is exact in this limit. The discrepancies between RPMD and QM results deserve a more careful analysis, which will be presented in Discussion section. The ICVT/µOMT rate coecients,

50

calculated in ref 36 and also shown in Figure 4,

are always below the RPMD results even at the highest temperatures.

It seems that the

ICVT/µOMT cannot account for tunneling to same extent as RPMD does. Given the small

◦ skewing angle for the present reaction (23.8 ), an appreciable part of the tunneling is likely to proceed along paths very much distorted from the MEP in a corner-cutting fashion, similarly to what has been found in other cases with comparable skewing angles such as the D+HMu exchange reaction.

10

As already shown, the QCT results deviate strongly from the QM and RPMD results, and the discrepancy increases with decreasing temperature. Lower panel of Figure 4 shows that QCT results are reliable only in the high-temperature limit (>1500 K) where quantum mechanical eects of nuclear motions, such as the ZPE or tunneling, have no eect. The experimental measurements by Singleton

et al. 53

in the high-temperature limit also

exhibit a strong deviation from the Arrhenius behavior at duced by any of the theoretical approaches.

18


T >2000 K

that cannot be repro-

Page 19 of 35

3 0

4 0 0 1 0

J = 0 1

J = 2 0 1

1 0

1 0

-1

1 0

-3

1 0

-5

3 0 0

1 0

-1

1 0

-3

1 0

-5

2 0 1 5

2 0 0

C

R

J

(E

to t

)

2 5

0 .4

0 .3

0 .5

0 .6

0 .7

0 .8

0 .4

0 .3

0 .5

0 .6

0 .7

0 .8

1 0 1 0 0

Q M 5

3

Q C T 0

A '

0 0 .4

0 .6 1 0

0 .8

1 .0

1 .2

1 .4

1 .6

J = 5 0 1

0 .6

0 .4

1 0

0 .8

1 .0

1 .2

1 .4

1 .6

J = 8 0 1

2 0 0

2 0 0

-1

1 0

-3

1 0

-5

1 0

-1

1 0

-3

J

(E

to t

)

1 0

R

C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 0 0

0 .5

0 .4

0 .6

0 .7

1 0 0

0 .8

0

0 .6

0 .5

0 .7

0 .8

0 .9

1 .0

0

0 .4

0 .6

0 .8

E

1 .0 to t

1 .2

1 .4

1 .6

0 .4

0 .6

0 .8

(e V )

1 .0

E

to t

1 .2

1 .4

1 .6

(e V )

Figure 5: Comparison of QM (black solid line with open circles) and QCT (red line) cumula3 tive reaction probabilities resolved in J , as a function of the total energy for the O( P )+HCl 3 0 reaction on the A PES at the indicated values of J . In each case, the inset depicts the low energy in logarithmic scale.

Discussion In spite of the high crossover temperature on both PESs, the limitation of the RPMD to reproduce the low temperature

k(T ) deserves some more detailed attention.

the QM results reproduce the experimental data up to

T = 1000 K

Moreover, while

very accurately, it is not

until 700 K that a similar degree of agreement is attained with the RPMD results. It is thus worth examining the CRPs for dierent

Js

and to compare QM and QCT results.

Figure 5 displays the comparison between QM and QCT CRPs on the 50, 80.

3

A0

for

J =0,

20,

In both QM and QCT results, the eective threshold increases with increasing

J

due to the centrifugal barrier. Although QCT CRPs reproduce the overall shape of the QM

19



ones, the dierences in the low energy region (highlighted in the respective insets) are evident and justify the much lower

k(T )

for the QCT results, as shown in Figure 3. Some smooth

resonance structure can be observed for

J =0,

but it practically disappears with increasing

J. 4 0

6 0 0 1 0

J = 0 0

1 0

-2

1 0

-4

1 0

-6

1 0

-8

to t

(E

1 0

-1

1 0

-3

1 0

-5

4 0 0 3 0 0

R

J

2 0

J = 2 0 1

1 0

5 0 0

)

3 0

0 .3

0 .4

0 .5

0 .6

0 .7

0 .4

0 .3

0 .5

0 .6

0 .7

C

0 .2

2 0 0 1 0

Q M

1 0 0

3

Q C T 0

A "

0 0 .4

0 .6

5 0 0

1 0

4 0 0

1 0

-1

1 0

-3

3 0 0

1 0

-5

0 .8

1 .0

1 .2

1 .4

1 .6 4 0 0

J = 5 0 1

0 .6

0 .4 1 0

0 .8

1 .0

1 .2

-1

1 0

-3

1 0

-5

1 .6

J = 8 0 1

1 0

1 .4

2 0 0

J

(E

to t

)

3 0 0

R

C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 35

2 0 0

0 .4

0 .3

0 .5

0 .6

0 .7

0 .4

0 .3

0 .5

0 .6

0 .7

0 .8

1 0 0

1 0 0 0

0

0 .4

0 .6

0 .8

1 .0

E

to t

1 .2

1 .4

1 .6

0 .4

0 .6

0 .8

(e V )

1 .0

E

Figure 6: Same as Figure 5 for the

The QCT and QM CRPs on the

3

A00

3

A00

to t

1 .2

1 .4

1 .6

(e V )

PES's.

PES are shown in Figure 6.

The QCT results

are able to reproduce the overall shape of the QM CRPs fairly accurately, but, as can be inspected in the respective insets, the QM CRPs extends at lower energies than the QCT ones.

In any case, apart from the obvious dierences at the lowest energies, the main

dierence between QCT and QM is the resonance structure in the latter that manifests as oscillations, sharper than on the

3

A0

PES, and that they are expected to promote tunneling.

Moreover, oscillations due to resonances are clearly seen in

20


CRJ (Etot ) for J 's as high as J =50.

Page 21 of 35

To emphasize the resonance structure for the reaction on the

3

A00

PES, Figure 7 displays the

density of reactive states (DORS); that is, the derivative of CRP with respect to the total energy.

57

As can be seen, there is a dense succession of quantized transition states. Usually,

with increasing

J,

the DORS peak structure smooths out. However, as can be seen in the

lower panel of Figure 7, the series of very sharp peaks is preserved for values. In contrast, on the

3

A0

J =20

and even larger

PES the DORS (not shown) is much smoother for

oscillations are almost absent for large

J 's

J =0

and

values.

D e n s ity o f r e a c tiv e s ta te s

1 0 0

J = 0

8 0

3

A ''

6 0 4 0 2 0 0 -2 0 0 .4

0 .6

0 .8

1 .0

E

1 5 0 0

D e n s ity o f r e a c tiv e s ta te s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


to t

1 .2

1 .4

1 .6

1 .2

1 .4

1 .6

(e V )

J = 2 0 3

A ''

1 0 0 0

5 0 0

0 0 .6

0 .4

0 .8

1 .0

E

to t

(e V )

Figure 7: Derivative of the CRP with respect to Etot (density of reactive states; DORS) on 3 00 A PES at J =0 (upper panel) and J=20 (bottom panel)

As shown in Figure 1, the

3

A00

exhibits vdW wells in the reactant and product channels,

before and after the barrier, which give rise to the formation of quasibound states hence leading to resonances by which the reactivity is greatly enhanced. The eect of the vdW

21



minima was examined in detail in a previous work prequel of the present

3

A00

35,58

with calculations on the S4 PES,

35

3

A0

a

at the low energy region shows that tunneling

through the combined electronic and centrifugal barrier is enhanced on the respect to the

38,58

PES, which also exhibits vdW wells. A detailed inspection of QM

CRPs calculated in this and previous works

3

A00

PES with

PES. This energy region, which corresponds to the deep tunneling regime

where multiple resonances take place, extends well below the QCT thresholds even for high

J

values. It can be easily shown that the QM rate coecients for specic

J

values,

k J (T ) are

between two and three orders of magnitude larger than the respective QCT in the 200-500 K. 1 .4 3

3

3 .0

A '

A "

1 .2

0 K

3 .0

3 .5

0 K 0 K 0 K

1 .0

s

-1

e v

-1

)

2 .5

1 0 0 2 0 0 3 0 0 3 5 0

0 .8 1 .5 0 .6 1 .0

0 .4

1 1

T h e rm a l C R P (c m

3

2 .0

1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 35

0 .5

0 .2

0 .0

0 .0 0 .0

0 .5

1 .0

1 .5

E

2 .0 to t

2 .5

3 .0

3 .5

4 .0

/ e V

0 .0

0 .5

1 .0

1 .5

E

2 .0 to t

2 .5

4 .0

/ e V

3 Figure 8: QCT thermal cumulative reaction probabilities (T -CRP) for the O( P )+HCl re3 0 3 00 action on the A (left panel) and A (right panel) PES's at 1000 K (black solid), 2000 K (red dash-dot), 3000 K (green dash), and 3500 K (blue solid).

Note the dierent ordinate

scale for the two PESs.

It is thus straightforward to explain the dierences found in the QM and RPMD rate coecients. The eect of the resonances originated by the vdW wells plays a crucial role in the higher values of the QM

k(T ),

even at moderate temperatures (≤

500 K).

Although

RPMD accounts fairly well for tunneling, which is a short-time eect, it does not when tunneling is enhanced by resonances, which is a long-time eect as is the case of the title reaction on the

3

A00

PES.

It remains to explain the discrepancies found at the higher temperatures where RPMD

22


Page 23 of 35

Q M 6 0

Q M

X ie e t a l. jm a x = 2 5

Q M

jm

a x

= 9 9

R

(E

to t

)

J = 0

4 0

C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 0

3

A "

0 0 .5

1 .0

1 .5

E Figure 9:

to t

2 .0

2 .5

(e V )

J = 0 Cumulative reaction probability (CRP) on the 3 A00 PES. Present with jmax = 25 (dot blue line), jmax = 99 (red solid line) are compared with the

QM

calculations

data from ref 36 (green dash-dot line)

k(T )

are somewhat larger than the presumably more accurate QM calculations from ref 36.

The highest total energy for these calculations was 2.67 eV. Figure 8 shows the QCT thermal CRPs on the

3

A0

(left panel) and

As can be seen, a maximum 2000 K but

max >3.0 Etot

3

A00

(right panel) at 1000 K, 2000 K, 3000 K and 3500 K.

max Etot = 2.75 eV is sucient to determine rate coecients at T ≤

eV is required to calculate

k(T = 3000 K).

Moreover, the QM calculations from ref 36 were performed with a rotational basis with

jmax =25.

However, this number of rotational states included in the calculations seems to

be insucient at high enough collision energies. To check this eect in the CRPs, we have calculated the

J =0

QM CRP for with two dierent

of the resulting CRPs is depicted in Figure 9. with

jmax =25

jmax

are indistinguishable for

jmax

values: 25 and 99. The comparison

In addition, the results of by Xie

et al. 36

are also shown in the gure. As can be seen, the plotted data with the same

Etot 1.6

eV, it becomes

conrming that, indeed, the latter rotational

basis is too short to account for accurate results at the total energies necessary to calculate

23



k(T )

Page 24 of 35

above 1500 K.

Putting together these limitations in the QM calculations from ref 36, the nding that at temperatures above 2000 K the RPMD renders larger

k(T )

than the QM calculations

becomes easily explainable. It should be stressed, however, that in spite of the above lim-

J -shifting

itations plus those that may result from the application of the

approximation

(although with sensible corrections), the QM results do not deviate more than 30% in the 2000-3200 K range. The rapid rising of tanovic

53

k(T ) found in the experiments by Singleton and Cve-

cannot be explained with any of the approaches using in this or previous works.

The discrepancy could be attributed to limitations in the PES, which does not account for the OCl+H product channel and may not describe the high energy region with enough ac-

o The H-elimination channel, which is highly endothermic (∆r H0

curacy.

≈

1.69 eV), was

studied at hyperthermal energies in crossed molecular beam experiments and `on the y' QCT calculations on a B3LYP density functional PES by Zhang

et al. 59

A threshold of

eV was found experimentally, coincident with that from the QCT calculations on a single PES. Subsequent QCT calculations on more rened interpolated

A0

and

A00

PESs by Binder

et al. 60

≈2 3

A00

ab initio (CASSCF-MRCI)

gave a similar threshold. In both studies, it was found

that at collision energies above 3.5 eV the H-atom elimination started to prevail over the H-atom abstraction, reaching a OCl/OH branching ratio of ingly, the latter study found that the contribution of the was comparable or even larger than that from the

3

A00 ,

3

≈3

A0

at

Ecoll =5

eV.

60

Interest-

PES to H-atom elimination

in contrast to what occurs with OH

formation. Since the rate coecients of the experiments of Singleton and Cvetanovic were determined by measuring the decay of O atoms under pseudo-rst order conditions,

53

it

could be envisaged that the opening of the OCl channel could be reason for the sudden rise of

k(T ).

However, a simple (and possibly generous) estimate of

function given in ref 60 leads to a value of

≈ 1 − 4 × 10−21

cm

3

≈ 1 − 3 × 10−14

cm

k(T ) 3

using the excitation

−1 s at

T =3000 K

and of

−1 s at 1000 K for OCl formation, practically negligible as compared to

the experimental and theoretical values of

k(T )

24

for OH formation at those temperatures.


Page 25 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The discrepancies with the high-T experimental results can be also attributed to the neglect of Renner-Teller coupling between the coupling.

3

A0

and

3

A00

PESs or the neglect of spin-orbit

The experiments of ref 53 were initiated by mercury-photosensitization of N2 O

to produce O(

3

P)

1 atoms. However, N2 O can also and mainly produce O( D ) although at

presumably shorter wavelengths. Therefore, a small fraction of O(

1

D), whose reactivity with

HCl is orders of magnitude larger than with the ground state oxygen atoms, cannot be ruled out completely, possibly leading to an overestimation of the rate coecient.

Conclusions This work was aimed to test the performance of the Ring Polymer Molecular Dynamics to calculate rate coecients for the O(

3

P)

+ HCl

→

OH + H in an ample range of tempera-

tures, 2003300 K. Accurate calculations for this reaction are challenging. First, they require calculations on two concurrent PESs,

3

A0

and

3

A00 , both with high energy barriers.

the former is much less reactive, its contribution cannot be neglected at ondly, the

3

A00

Although

T >1500 K.

Sec-

exhibits a bent transition state that together with a small skewing angle

and a high crossover temperature, 420 K, causes the tunneling contribution to be decisive at temperatures as high as 500 K. Finally, the presence of vdW wells gives rise to a resonance structure that enhances tunneling. many partial waves (J

>200),

Exact QM calculations require the inclusion of

large rotational basis,

jmax ≈60 and a considerable number of

helicity projections. For these reasons, RPMD appears as an alternative to the much more expensive QM calculations. The present results show that RPMD predictions underestimate

k(T )

between a factor from 6 to 2 in the range 300500 K, but at

is very good.

T >700 K

the agreement

In contrast, QCT calculations or even ICVT/µOMT fall below the RPMD

results in all but the highest temperature range. In addition to the RPMD calculations, QCT and QM cumulative reaction probabilities have been calculated for several total angular momenta. Comparison of the respective results

25



shows that the overall shape, especially at high energies, is well reproduced by QCT results. However, the large dierences in the

k(T )

values at lower temperatures are due to the high

classical thresholds. It is precisely the dominant role of tunneling boosted by the presence of resonances that makes the RPMD

k(T )

to be smaller than the QM ones below 500 K.

Finally, a detailed examination has shown that limitations in QM calculations of ref 36 (too small values of

max Etot ,

and above all of

jmax )

give rise to an underestimation of the rate

coecients at temperatures above 2000 K with respect to the RPMD ones. In summary, it is shown that even for the title reaction RPMD constitutes an alternative to accurate QM calculations, much more computationally demanding. Due to the presence of resonances and other dynamical features, the O(

3

P )+HCl

is indeed a very stringent test for

the the RPMD method, which, except for the lowest temperatures, is able to provide reliable results in a wide range of temperatures. In addition, new fully converged QM calculations of rate coecients in a wide range of temperatures on an accurate global PES that includes the OCl channel are indeed desirable. These calculations would be a milestone in accurate calculations of bimolecular reactions and would imply a

tour de force as far as the necessary

computational eort is concerned.

Acknowledgement We are grateful to Prof. B. Ramachandran for kindly providing us with the Fortran codes of the two potential energy surfaces used in this work. Funding by the Spanish Ministry of Science and Innovation (grants MINECO/FEDERCTQ2015-65033-P, and PGC2018-096444B-I00) are acknowledged. P.G.J. also acknowledges funding by Fundación Salamanca City of Culture and Knowledge (programme for attracting scientic talent to Salamanca). Y.V.S. thanks the European Regional Development Fund and the Republic of Cyprus for support through the Research Promotion Foundation (Projects: INFRASTRUCTURE/1216/0070 and Cy-Tera NEA

YΠO∆OMH

/

ΣTPATH/0308/31)

26

.


Page 26 of 35

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Page 35 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Graphical TOC Entry (3P) +

35


New Stress Test for Ring Polymer Molecular Dynamics: Rate

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