Collision Efficiency Parameter Influence on Pressure-Dependent Rate Constant Calculations Using the SS-QRRK Theory

The system-specific quantum Rice–Ramsperger–Kassel (SS-QRRK) theory (J. Am. Chem. Soc.2016, 138, 2690) is suitable to determine rate constants below the high-pressure limit. Its current implementation allows incorporating variational effects, multidimensional tunneling, and multistructural torsional anharmonicity in rate constant calculations. Master equation solvers offer a more rigorous approach to compute pressure-dependent rate constants, but several implementations available in the literature do not incorporate the aforementioned effects. However, the SS-QRRK theory coupled with a formulation of the modified strong collision model underestimates the value of unimolecular pressure-dependent rate constants in the high-temperature regime for reactions involving large molecules. This underestimation is a consequence of the definition for collision efficiency, which is part of the energy transfer model. Selection of the energy transfer model and its parameters constitutes a common issue in pressure-dependent calculations. To overcome this underestimation problem, we evaluated and implemented in a bespoke Python code two alternative definitions for the collision efficiency using the SS-QRRK theory and tested their performance by comparing the pressure-dependent rate constants with the Rice–Ramsperger–Kassel–Marcus/Master Equation (RRKM/ME) results. The modeled systems were the tautomerization of propen-2-ol and the decomposition of 1-propyl, 1-butyl, and 1-pentyl radicals. One of the tested definitions, which Dean et al. explicitly derived (Z. Phys. Chem.2000, 214, 1533), corrected the underestimation of the pressure-dependent rate constants and, in addition, qualitatively reproduced the trend of RRKM/ME data. Therefore, the used SS-QRRK theory with accurate definitions for the collision efficiency can yield results that are in agreement with those from more sophisticated methodologies such as RRKM/ME.


Derivation of the Gilbert et al. 1 Correction Factor ( ) ∆
The general equation is given by the integral: Re-writing the exponential in brackets: Solving the brackets: Normalizing by : Which is the final expression of the complementary factor.

Python Code for the Implementation of the Correction Factors within the SS-QRRK/MSC ∆ Approaches
In the following code are presented the data for 1-pentyl radical decomposition. The file C5_SI.xlsx is given as a reference. In the sheet "Rates" is specified the high-pressure limit (HPL) rate constant of the unimolecular reaction and in the sheet "Energy" one types "0" if the reaction is endothermic or "1" if the reaction is exothermic. In the sheet "SS-QRRK", data and 22 are used to calculate the energy _ transferred in the deactivation process as . and specify the 〈 〉 = _ • ( /300) Lennard-Jones parameters ( , and MW) of the reactant and bath gas, respectively. For the bath gas, / , the two critical parameters ( and ) are also requested to solve the Redlich-Kwong equation of state.
Column P specifies the pressure values for the calculations. The frequencies of the reactant is includen in the colum and its zero-point energy in the column . The user has to specify the name of the Excel _ file in line 24, the column name of the HPL rate constants in line 25, and the cell of the endothermicity or exothermicity in line 26 (which is located in the sheet "Energy"). Alternatively, the user can modify the line 53 of the code related with the number of iterations and the line 91 referred with the initial parameters S3 to obtain the Tolman activation energies and the pre-exponential factors used in SS-QRRK/MSC computations.

Pressure Dependent Rate Constants with Nitrogen as the Bath Gas
Pressure in atm and rate constants in s -1 .

Pressure Dependent Rate Constants with Argon as the Bath Gas
Pressure in atm and rate constants in s -1 .     Figure S3. Collision efficiency parameter ( ) calculated for the unimolecular tautomerization with argon as bath gas using the three different approaches described in this work.

Rate Constants for the Propen-2-ol Unimolecular Tautomerization Computed by the Different SS-QRRK/MSC Approaches and The RRKM/ME Method
Pressure in atm and rate constants in s -1 .

Pressure Dependent Rate Constants for Hydrocarbons Decomposition Computed with the Approaches Proposed in this Work
Pressure in atm and rate constants in s -1 .