Three-Dimensional Master Equation (3DME) Approach - The Journal

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Three-Dimensional Master Equation (3DME) Approach Thanh Lam NGUYEN, and John Stanton J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b06593 • Publication Date (Web): 04 Sep 2018 Downloaded from http://pubs.acs.org on September 4, 2018

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The Journal of Physical Chemistry

Three-Dimensional Master Equation (3DME) Approach Thanh Lam Nguyen1 and John F. Stanton1,* 1

Quantum Theory Project, Department of Chemistry and Physics, University of Florida, Gainesville, FL 32611 (USA).

Corresponding author: [email protected]

Abstract: The master equation technique is a standard tool to interpret gas-phase experimental kinetic results as well as to provide phenomenological rate coefficients for modeling. When there are significant changes of rotational constants along the reaction coordinate from a reactant through a transition state (TS) to product(s), including effects of angular momentum explicitly into a master equation model becomes vitally important. In this work, assuming that the K quantum number is adiabatic for both TS and reactant, we have developed an algorithm for pragmatic solutions of a three-dimensional master equation (3DME) that involves internal energy, total angular momentum (J) and its projection K. Two examples (one is for a thermally activated isomerization of CH3NC to CH3CN via a tight TS; and the other is for a thermally activated dissociation of NH3 to H + NH2 via a loose, variational TS) are given. In addition, comparison of 3DME results with experiment as well as with those of 1DME and 2DME are documented.

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I.INTRODUCTION Chemical reactions occurring in combustion, atmospheric, and interstellar environments are very complicated and often include hundreds or even thousands of elementary reactions that can occur consecutively and/or in parallel.1-3 Knowledge of mechanisms and kinetics for these elementary reactions is important because such information is required for modeling in order to better understand the whole process. Both advanced experimental techniques4-7 and high-level theoretical calculations8-10 can be used to obtain reliable reaction rate coefficients, which are given as functions of both pressure and temperature. However, experimental results are limited since practical considerations dictate that they can be measured only in a limited range of temperature and pressure. Master equation techniques11-31 are useful aids to interpret and complement experimental results, and to make independent predictions when experimental results are lacking. A master equation includes a system of intergo-differential equations that describe a highly competitive process of unimolecular reactions (i.e. isomerization or dissociation) with collisional energy transfer procesess.26-28 Solutions of a master equation yield reaction rate coefficients and product branching ratios calculated as functions of both temperature and pressure.26-28 For the one-dimensional master-equation (where only internal energy is considered; effects of angular momentum are either neglected or included approximately), the solutions are well documented through both deterministic (i.e. matrix) and stochastic methods.11-28 In addition, several chemical kinetics software packages are available.26-28 For some cases (for example, dissociations with a loose, variational TS such as CH4 → H + CH3,8 O3 ↔ O2 + O,32-33 NH3 → NH2 + H, so on) where there are significant changes of rotational


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constants along the reaction coordinate, including effects of angular momentum explicitly into the master equation model may be a necessity.34-35 While the total angular momentum quantum number J (a good quantum number) is always conserved, its projection (the K quantum number–not a good quantum number for asymmetric tops) is not.36-41 Therefore, the convolution of external rotational quantum states with vibrational quantum states depends on how K is treated: as either adiabatic or active. Because of this, there are, in principle, four different models that can be used to compute microcanonical rate constants:36-41 (i) first, model-I assumes that K is active for both TS and reactant; (ii) second, model-II uses K adiabatic for both TS and reactant; (iii) third, model-III applies K active to TS, but K adiabatic to reactant; and (iv) final, model-IV assigns K adiabatic to TS, but K active to reactant. Model-I with K active36-37 for both TS and reactant is very popular and also called the Jshifting approximation42-45 used in quantum dynamics calculations. In model-I, it is assumed that (three) external rotations can exchange energies freely and statistically with the vibrations; thus, sums and densities of rovibrational states can be calculated as:42-45

(, ) = − (, ) (1)

(, ) = − (, ) (2)

(, ) = ( + 1) + ( − ) , !ℎ = √ ∙ % &'( || ≤ (3) According to RRKM theory,46 the microcanonical (E,J)-resolved rate constant is then given by: . (, - ) ,(, ) = ∙ (4) ℎ (, )


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Where A, B, and C are three external rotational constants for a polyatomic molecule with B ≈ C, σ is the reaction path degeneracy, h is Planck’s constant, and Erot (in eq. 3) represents rotational energy levels of a rigid-rotor symmetric top molecule. As a result, model-I leads to a formalism for a two-dimensional master equation (2DME), whose (approximate) solutions for a one-well reaction system can be found in the literature.8,

15, 23, 31, 47-48

Unlike model-I, the remaining

models (II to IV) require a three-dimensional master equation (3DME) approach. Recently, Jasper et. al.8 have published a paper on a full solution for the twodimensional master equation where the (E,J)-resolved collisional transfer rates were determined through trajectory calculations. The pressure-dependent rate coefficients calculated for the unimolecular dissociation reactions CH4 → CH3 + H and C2H3 → C2H2 + H generally agree (within 20%) with experiment.8 Since such trajectory calculations are expensive, they have not been widely applied. In practical applications, the 2DME is commonly simplified to 1DME, which can be solved routinely.21, 49-55 Various simplified models were suggested and have been extensively reviewed.21, 49-55 It is well established that all simplified (1DME) models give an exact result at the high-pressure limit, but not at the low-pressure limit.21,

49-55

To

overcome this limitation, we have recently proposed to use a fixed-J model,56-59 which will always give an exact result at the low-pressure limit. In addition, the fixed-J approximation can also give an exact result at the high-pressure limit,51, 60 and was first recommended by Forst.51, 60

By assuming a weak-collisional internal energy relaxation/fixed-J model, such a 2DME can be

reduced to a number of 1DMEs, whose solutions (for an arbitrary complicated reaction system having multiple intermediates and multiple products) can be calculated very efficiently in parallel (viz. coarse-grained parallelism).10, 56-59 Like other 1DME models, the fixed-J approach


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has a drawback that does not take the fully-coupled (E,J)-resolved collisional transfer rates into account. However, this approach seems to work as long as the amount of total angular momentum (γ) transferred per collision in a downward direction is (negligibly) small, as is indeed observed (e.g. γ ≤ 2 in Fig. 1) in the recent work.8 In other words, the collisional transfer rates are assumed to depend only on an initial angular momentum state, but not (or weakly) on a final angular momentum state. Given this, it may be worthwhile to consider a model in which both J and K are fixed, as we do in this work. It should be mentioned that, for a polyatomic symmetric top molecule with a rigid-rotor model as used here, the K quantum number (having an index number with limits of –J to +J) is also conserved,42 thus resulting in a three-dimensional master equation. In practice, however, most stationary points (especially for TSs) are asymmetric tops, so the K quantum number is not conserved. In such a case, K is an index number listing the 2J + 1 asymmetric top rotational energy levels corresponding to total angular momentum J. As a result, a 2DME approach may be sufficient. However, to obtain rotational-(vibrational) eigenvalues for such stationary points, somewhat involved numerical calculations are required. In this work, we will develop an algorithm to again obtain pragmatic solutions of a three-dimensional master equation. We will use model-II, which assumes the K quantum number is adiabatic for both the TS and reactant.37 Model-II has recently been found to give good results (as compared to trajectory calculations) for the O2 + O ↔ O3 reaction.32-33, 61 For the isomerization system of CH3NC → CH3CN,59 modelII was also found to give better results than the other two models (III and IV) involving an adiabatic treatment of K (see the Supporting Information in ref. 59).


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This paper is organized as follows: section II describes a practical 3DME approach and its solutions; two examples (one is for a thermally activated isomerization of CH3NC to CH3CN via a tight TS; the other is for a thermally activated dissociation of NH3 to H + NH2 via a loose, variational TS) will be discussed in section III. In addition, comparison of 3DME results with those of 1DME and 2DME is also included here. Finally, section IV summarizes the important results.

II. THREE-DIMENSIONAL MASTER EQUATION APPROACH Assuming that all stationary points are rigid-rotors and symmetric tops, both J and K quantum numbers are adiabatic and conserved along the reaction coordinate from a reactant via a TS to product(s), and the microcanonical (E,J,K)-resolved rate coefficient can be expressed as:37 ,(, , ) =

- . ( . , , ) ∙ (5) ℎ (, , )

Where sums of (E,J,K)-resolved states for a TS and densities of (E,J,K)-resolved states for a reactant are calculated as:37

. (, . ( . , , ) = . − ) (6)

(, , ) = − (, ) (7)

For a thermally activated reaction system including one well and one product (e.g. → 4), a 3DME model that describes time-evolution of populations of the intermediate and the product at particular conditions of temperature and pressure is given in Eq. (8):4, 6, 21


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; = =

5%(6 , 6 , 6 ) =

5!

;

7 4(6 , 6 , 6 |8 , 8 , 8 ) ∙ 9: ∙ %(8 , 8 , 8 )(8 − 9:

= > = =

∙ %(6 , 6 , 6 ) − ,(6 , 6 , 6 ) ∙ %(6 , 6 , 6 ) (8)

Where ωLJ (in s−1) is the collision frequency, C is the population of quantum states for the

energized molecule, and 4(6 , 6 , 6 |8 , 8 , 8 ) is an energy/angular momentum transfer probability function from an initial state (Ek,Jk,Kk) to a final state (El,Jl,Kl). Eq. (8) has an analytical solution at the high-pressure limit where the thermal rate constant can be computed using transition state theory (TST):62-64 ;

;

,(@)A; = 7 ,(, , ) ∙ BC (, , )( (9) >

Where FB is Boltzmann’s thermal energy distribution function, which is expressed as: BC (, , ) =

(2 + 1)(, , )exp (−

) H@

∑; > ∑ J(2 + 1)(, , )exp (− ;

H@)(

(10)

For a given value of J, there is a (2J + 1)–fold degeneracy. It should be noted that the (2J+1)

factor in Eq. (10) is not included in the computed (, , ) in Eq. (7).

At pressures in the falloff region, one must solve Eq. (8) numerically, provided that 4(6 , 6 , 6 |8 , 8 , 8 ) is identified beforehand. Unfortunately, 4(6 , 6 , 6 |8 , 8 , 8 ) is very

poorly known. In principle, it can be obtained from either classical trajectory calculations or highly (E,J,K)-resolved experimental measurements.7 But, such works are extremely difficult to do, and thus rare. Even a simpler formalism of the two-dimensional function 4(6 , 6 |8 , 8 ) is

not determined. Some studies done recently by Barker et. al.65-66 and by Conte et. al.67 that


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tried to identify a feasible P function have been reported in the literature,65-66 but further work is clearly needed. Because of these reasons, to obtain pragmatic solutions for Eq. (8) in this work we have assumed that the population distribution depends parametrically on the initial (8 , 8 ) state (i.e. before collision), but not on the final (6 , 6 ) state (i.e. after collision). Strictly

speaking, total internal energy is allowed to change by collisions, but the angular momentum J and K quantum numbers change very slightly (or remain unchanged) by collisions. Such a simple assumption is only valid at two extreme conditions: at the low- and high-pressure limits, but not at other pressures in the falloff region. Particularly, we have used a weak-collisional internal energy relaxation/fixed (J,K) angular momentum model to solve Eq. (8). It should be mentioned that this assumption is analogous to the 1DME approach, where only internal energy is allowed to change when an energized molecule collides with the bath gas. Following the general strategy used in our 2DME model (where angular momentum quantum numbers are kept constant and considered as conventional parameters in numerical calculations), the 3DME in Eq. (8) can be reduced to a number of 1DMEs, Eq. (11), whose solutions are well established.11-31 ;

5%(6 ), = 7 4(6 |8 ), ∙ 9: ∙ %(8 ), (8 − 9: ∙ %(6 ), − ,(6 ), ∙ %(6 ), (11&) 5!

In a formulation of the energy-grained master equation, Eq. (11a) becomes:4, 6, 21 5%(6 ), = 4(6 |8 ), ∙ 9: ∙ %(8 ), − 9: ∙ %(6 ), − ,(6 ), ∙ %(6 ), (11L) 5! ;

Eq. (11b) can then be cast to a matrix form, Eq. (11c):

5ℂ, = N, ∙ ℂ, (11O) 5!


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Here N, is the relaxation matrix, which has structure shown below: N,

S, (1,1) 9: 4(1|2) 9: 4(2|1) S, (2,2) R =Q ⋮ 9: 4(U − 1|1) 9: 4(U − 1|2) 9: 4(U|2) P 9: 4(U|1)

9: 4(1|U − 1) 9: 4(1|U) 9: 4(2|U − 1) 9: 4(2|U) Z ⋱ ⋮ Y S, (U − 1, U − 1) 9: 4(U − 1|U) ⋯ 9: 4(U|U − 1) S, (U, U) X ⋯

with S, ([, [) = −,(\ ), − 9: + 9: 4([|[)

(12)

In this work, a commonly used single exponential function13, 26 is chosen for the energy transfer probability in a downward direction, Pd; the upward energy transfer probability (Pu) is then determined by considerations of detailed balance:4, 6, 21 4] ([, ,) =

1 8 − \ ∙ exp ^− b !ℎ 8 ≥ 6 (13&) %8 〈`〉]

4d ([, ,) ∙ BC (,) = 4] (,, [) ∙ BC ([) !ℎ 8 < 6 (13L) Where FB is the Boltzmann energy distribution of the molecule, Ck is the normalization factor, and d is the average amount of energy transferred per collision in a downward direction. It should be mentioned that the exponential model used here neglects the well-known experimental finding (viz. d = f(E)) and is certainly inferior to high-accuracy models derived from trajectory calculations.8, 65, 67

Note that N, is an asymmetric matrix, which can be symmetrized to f, by the following transformation (according to the detailed balance, see Eq. (13) above):


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f, = gN, gh

(14a)

if = giN or iN = gh if

(14b)

Here if is an eigenvector of matrix f, , g is a diagonal transformation matrix, which has typical elements given by: j([, [) =

h

klm (\)

and j([, [)h = kBC ([)

(15)

In this work, at each (fixed) J and K, we diagonalize the symmetric matrix f, to obtain all eigenvalues (⋀) and corresponding eigenvectors (of ). Because of the transformation property, all eigenvalues of f, remain the same as those of N, ; but with different

eigenvectors. The eigenvectors of N, can then be computed from those of f, using Eq. 14b. The population distribution of the energized molecule (at each fixed J and K) as a function of time can subsequently be calculated using Eq. 16; and the phenomenological rate constant is equivalent to the magnitude of the smallest eigenvalue (|λ1|), Eq. 17. ℂ(!), = oN exp (⋀!)oh N ℂ(! = 0),

(16)

,dpq (@, 4), = −rh

(17)

Finally, the population of energized molecules and associated thermal rate constants at each temperature and pressure can be calculated as expectation values weighted by the Boltzmann thermal energy distribution and summed over all possible angular momenta (J and K) quantum numbers: 〈%(!)〉 =

stu stu ∑> ∑ J %(!), ∙ BC (, , )(
∑ J BC (, , )(
∑ J ,dpq (@, 4), ∙ BC (, , )(
∑ J BC (, , )(
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