Detailed kinetic mechanism for ch300 + no reaction an ab initio study

ABSTRACT Detailed kinetic mechanism of the CH3OO + NO reaction was intensively investigated using the accurate composite W1U method and Rice-Ramsperger-Kassel-Marcus/Master Equation RRKM

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

BACH KHOA UNIVERSITY



NGUYEN THI HOAI

DETAILED KINETIC MECHANISM FOR

Major: Chemical Engineering

Major ID: 60520301

MASTER THESIS

Ho Chi Minh City, August 2017

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

BACH KHOA UNIVERSITY



NGUYEN THI HOAI

DETAILED KINETIC MECHANISM FOR

MAJOR: CHEMICAL ENGINEERING

MAJOR ID: 60520301

MASTER THESIS

Ho Chi Minh City, August 2017

CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA–ĐHQG–HCM

Cán bộ hướng dẫn khoa học : PGS.TS Huỳnh Kim Lâm

Cán bộ chấm nhận xét 1: PGS TS Lê Thị Lý

Cán bộ chấm nhận xét 2: TS Phạm Hồ Mỹ Phương

Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG TP HCM ngày 01 tháng 08 năm 2017

Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:

1 PGS TS Phạm Thành Quân

2 PGS TS Lê Thị Lý

3 TS Phạm Hồ Mỹ Phương

4 PGS TS Nguyễn Quốc Thiết

5 TS Lê Xuân Tiến

Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý

chuyên ngành sau khi luận văn đã được sửa chữa (nếu có)

CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA KỸ THUẬT HÓA HỌC

ĐẠI HỌC QUỐC GIA TP.HCM

TRƯỜNG ĐẠI HỌC BÁCH KHOA

CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự do - Hạnh phúc

NHIỆM VỤ LUẬN VĂN THẠC SĨ

Họ tên học viên: Nguyễn Thị Hoài MSHV: 7140191 Ngày, tháng, năm sinh: 19/08/1991 Nơi sinh: Đăk Lăk Chuyên ngành: Kỹ thuật Hóa học Mã số : 60520301

I TÊN ĐỀ TÀI: Nghiên cứu cơ chế động học chi tiết của phản ứng CH3OO + NO bằng phương pháp tính toán cấu trúc lượng tử và thống kê

II NHIỆM VỤ VÀ NỘI DUNG:

1 Xây dựng bề mặt thế năng của phản ứng CH3OO + NO

2 Tính toán các thông số nhiệt động của chất phản ứng, sản phẩm trung gian, trạng

thái chuyển tiếp và sản phẩm phản ứng

3 Tính toán hằng số tốc độ phản ứng theo nhiệt độ và áp suất

III NGÀY GIAO NHIỆM VỤ : 11/01/2016

IV NGÀY HOÀN THÀNH NHIỆM VỤ: 01/08/2017

V CÁN BỘ HƯỚNG DẪN : PGS.TS Huỳnh Kim Lâm

Tp HCM, ngày 17 tháng 8 năm 2017

CÁN BỘ HƯỚNG DẪN

(Họ tên và chữ ký) CHỦ NHIỆM BỘ MÔN ĐÀO TẠO (Họ tên và chữ ký)

TRƯỞNG KHOA KỸ THUẬT HÓA HỌC

(Họ tên và chữ ký)

ACKNOWLEDGEMENTS

First and foremost, I would like to express my deepest gratitude to my supervisor, Assoc Prof Huynh Kim Lam, for his excellent guidance and valuable discussions for nearly three years

This thesis would not have been possible without the guidance and the help of Mai Van Thanh Tam and Duong Van Minh from the Institute for Computational Science and Technology–HCMC Special thanks for their kindness, patience and encouragement so that I can complete this study

I express my thanks to all my lecturers of the School of Chemical Engineering at Bach Khoa University for their amazing courses and their help

I would like to thank the Institute for Computational Science and Technology–HCMC for computing resources and financial support and International University, VNU–HCMC for gratefully acknowledged

I would also like to show my gratitude to Prof M.C Lin I am grateful to him for his kindness, his help and valuable advice during my internship period in Taiwan Last but not the least, I would like to thank my parents, my older sisters, and my friends, who were always supporting me and encouraging me with their best wishes

ABSTRACT

Detailed kinetic mechanism of the CH3OO + NO reaction was intensively investigated using the accurate composite W1U method and Rice-Ramsperger-Kassel-Marcus/Master Equation (RRKM/ME) statistical rate models The temperature- and pressure-dependent behaviors, including the time-resolved species profiles as well as rate coefficients of the elementary reactions, were rigorously characterized in a wide range of conditions (T = 298–1500 K & P = 7.6–76000 Torr) The computed thermodynamic data and rate coefficients are in good agreement with literature data in which the CH3O + NO2 was found to be the most dominant product channel The detailed kinetic mechanism, consisting of all calculated thermodynamic data (in NASA polynominal format) and kinetic data (in modified Arrhenius format), was explicitly provided for further detailed kinetic modeling and simulation of NOx-related applications in atmosphere and combustion

là sản phẩm chính ở điều kiện khảo sát Cơ chế động học chi tiết, bao gồm các thông

số nhiệt động lực học theo định dạng đa thức NASA và động học theo định dạng Arrhenius, được cung cấp để người dùng có thể sử dụng trực tiếp cơ chế này cho việc

mô phỏng động học và mô phỏng chi tiết các ứng dụng liên quan đến NOx trong khí quyển và quá trình đốt cháy

TABLE OF CONTENTS

LIST OF FIGURES iii

LIST OF TABLES v

LIST OF ABBREVIATIONS vi

CHAPTER 1 INTRODUCTION 1

CHAPTER 2 THEORETICAL BACKGROUND 4

2.1 The time-independent Schrödinger equation and the Born-Oppenheimer approximation 4

2.2 Hartree-Fock theory 6

2.3 Coupled-cluster methods 8

2.4 Density-based methods 10

2.5 Composite methods 12

2.6 Basis sets 13

2.7 Standard statistical mechanic methods 14

2.8 Transition state theory 16

2.9 Pressure-dependence analysis 18

2.10 Rate constant extraction 21

CHAPTER 3 COMPUTATIONAL DETAILS 26

3.1 Electronic structure calculations 26

3.2 Thermodynamic property and rate constant calculations 26

CHAPTER 4 RESULTS AND DISCUSSION 29

4.1 The optimized geometries of all species on the potential energy surface of CH3OO + NO reaction 29

4.2 Potential energy surface 33

4.3 Thermodynamic properties 37

4.4 Kinetic calculations/analysis 41

4.4.1 Time-resolved species profiles 41

4.4.2 Rate constant calculations 43

4.4.3 Pressure- and temperature- dependent analysis 46

CHAPTER 5 CONCLUSIONS AND OUTLOOK 49

LIST OF PUBLICATIONS 51

REFERENCES 52

APPENDIX 59

CURRICULUM VITAE 71

Figure 4.2 The optimized geometries of the relevant intermediates on the potential

energy surface of CH3OO + NO reaction, calculated at B3LYP/cc-pVTZ+1d level of theory Bond lengths are given in Å 31

Figure 4.3 The optimized geometries of the relevant transition structures on the

potential energy surface of CH3OO + NO reaction, calculated at B3LYP/cc-pVTZ+1d level of theory Bond lengths are given in Å 32

Figure 4.4 Potential energy surface for the reaction of CH3OO + NO at 0 K, calculated at W1U level of theory Zero-point energy corrections were included Numbers are energies (in kcal·mol-1) relative to the entrance channel (CH3OO + NO) 34

Figure 4.5 Time-resolved species profiles for the reaction CH3OO + NO at the pressure of 760 Torr and two different temperatures (298 and 1500 K) 42

Figure 4.6 Time-resolved species profiles for the reaction CH3OO + NO at 298 K and two different pressures (100 and 760 Torr) 43

Figure 4.7 Comparison between calculated and experimental rate coefficients as a

function of temperature at different pressures Experimental data are from the works

of Bacak et al., Masaki et al., Scholtens et al., Xing et al The experimental studies

were carried out over the pressure range 4–200 Torr and the temperature range 193–

298 K 46

Figure 4.8 Rate coefficients for CH3OO + NO  products as a function of

temperature at 76, 7600 and 76000 Torr (a–c) and as a function of pressure at 300,

800 and 1500 K (d–f) Calculations used the exponential down model with <Edown>

= 100 cm-1 for adduct CH3OONO (L-J parameters are ε/kB = 484 K and σ = 4.3 Å) and He as collider (ε/kB = 10.0 K and σ = 2.55 Å) 48

LIST OF TABLES

Table 4.1 Relative zero-point corrected energies (at 0 K in kcal·mol-1)for selected species involved in the reaction 35

Table 4.2 Comparison of the calculated reaction barrier heights (at 0 K in kcal·mol

-1) with the available literature data 37

Table 4.3 Calculated thermochemical parameters for methylperoxy radical

(CH3OO) Numbers are in kcal·mol-1, otherwise, they will be noted 38

Table 4.4 Calculated thermodynamic properties of selected species involved in the

Thermochemical Tablesa, NIST = Webbook NISTb, and Burcat = Burcat’s thermodynamic databasec) Units: kcal·mol-1 for fH (298 K)and cal·mol-1·K-1 for S and Cp 39

Table 4.5 Calculated energetic data for selected dissociation reactions (in kcal·mol

-1) at the W1U level of theory comparing with available literature data 41

Table 4.6 Calculated high-pressure rate constantsa for selected reaction pathways of the title reaction 45

LIST OF ABBREVIATIONS

AOs Atomic spin-orbitals

B3LYP Becke 3-parameter Lee-Yang-Parr

CASSCF Complete active space multiconfiguration self-consistent field

CC Coupled-cluster

CCSD(T) coupled-cluster with single and double and perturbative triple

excitations

CI Configuration interaction

CSFs Configuration (state) functions

CTST Canonical transition state theory

GTO Gaussian-type orbitals

IRC Intrinsic reaction coordinate

MCSCF Multi-configurational self-consistent field

MOs Molecular spin-orbitals

MPn Møller-Plesset perturbation theory

SCF Self-consistent field procedure

STO Slater-type orbitals

TST Transition state theory

CHAPTER 1 INTRODUCTION

The reaction between methylperoxy radical CH3OO and nitric oxide NO plays

an important role in combustion and atmospheric chemistry since it provides insights into some practical processes (e.g., the ignition of the hydrocarbon fuels in engines and the production of ozone in the troposphere) Kinetic mechanism of this reaction

is also very interesting, in particular, due to the opportunity to include its kinetic data for the modeling of combustion in engines using exhaust-gas recirculation and the NO-to-NO2 conversion by hydrocarbon fuels in recent NOx-reduction strategies [1, 2] Success in NOx reduction strategies depends crucially on the understanding of the chemical reactions occurring on the engines during combustion processes Moreover,

in tropospheric chemistry, NO can react with CH3OO to convert to NO2, whose influence on the ozone production is well-known [3]

Due to its importance, the reaction between CH3OO and NO has been the subject of numerous theoretical and experimental studies [4-18] The mechanism of this reaction was studied at different levels of theory, namely single-reference wave-function based [12, 17, 18], density functional [12, 17], multi-reference [13] and composite methods [12, 18] The title reaction was suggested to occur mainly via two channels [19] as follows:

Arenas et al [13] suggested there is no TS for CH3OONO dissociation to both CH3O + NO2 and CH3OO + NO products The authors also proposed a two-step mechanism for the CH3OONO isomerization to CH3ONO2 However, this mechanism was suggested to occur only at very low pressure [17] The effect of halogenation on the mechanism of the reaction CH3OO + NO was studied by Kosmas et al [17] using ab

initio and density functional theory (DFT) methods These authors proposed a triplet

TS for the isomerization pathway between CH3OONO and CH3ONO2

Experimentally, Scholtens et al [5] suggested the overall CH3OO + NO rate constant of (7.8 ± 2.2) × 10-12 cm3·molecule-1·s-1 at room temperature and 100 Torr

pressure Xing et al [10] investigated this reaction using time-resolved negative

ionization mass spectrometry and reported a higher rate constant of (9.9 ± 0.1) × 10-12

cm3·molecule-1·s-1 Butkovskaya et al [16] examined the pressure and temperature

dependences of the methyl nitrate formation channel to suggest the negative and

positive dependence of the branching ratio β = k1b/k1a on temperature and pressure, respectively With a simplified calculated potential energy surface (PES), this system was studied with the focus on nitrate formation by the use of master equation calculations [7, 11] and by quasi-classical trajectory calculations [15]

In this thesis, we systematically explored the detailed mechanism of the

CH3OO + NO reaction using the accurate composite electronic-structure W1U method [20] The electronic structure results were then used to characterize the time-resolved temperature- and pressure-dependent behaviors within the Rice-Ramsperger-Kassel-Marcus/Master Equation (RRKM/ME) framework for a wide range of conditions relevant to both atmospheric and combustion applications The constructed kinetic mechanism, consisting of thermodynamic data in NASA polynominal format and kinetic data in modified Arrhenius format, of the title reaction was provided to advance NOx-reduction strategies for real applications

The thesis consists of five chapters Chapter 1 shortly reviews the reaction between CH3OO and NO Chapter 2 introduces the theoretical background in computational chemistry Chapter 3 presents the computational details, which were

carried out in the present thesis Chapter 4 presents the results of our theoretical study

on the kinetic mechanism of the CH3OO + NO reaction Finally, Chapter 5 summarizes the results of our work and provides an outlook for our future study

CHAPTER 2 THEORETICAL BACKGROUND

Computational chemistry is a branch of chemistry which is simply the application of chemical, mathematical, and computing skills to investigate structures and properties of interesting chemical compounds in different phases, instead of carrying out experiments with chemicals Computational chemistry methods can be classified into two categories: semi-empirical (also empirical) methods which use parameter sets derived from experimental data to provide the input into mathematical

models and ab initio methods Ab initio is a Latin term meaning ‘from the beginning’

or ‘from first principles’ Ab initio methods are based on the laws of quantum

mechanics to provide accurate chemical predictions for chemical systems that required explicit treatment of electronic structure

2.1 The time-independent Schrödinger equation and the Oppenheimer approximation

Born-Equivalent to the Newton’s law in classical mechanics, the fundamental equation of motion in quantum mechanics is the Schrödinger wave equation postulated in 1926 [21], the time-independent form of which is:

N I=1

n i=1

N I=1

N I=1

n j≠i

n i=1

(2.2)

where the first two terms are the kinetic energy of the nuclei and of the electrons, respectively, the third term is the nuclear-nuclear repulsion, the fourth one is the electron-electron interaction, and the last one is the electron-nuclear interaction; Ψ is

the wave function; and E is the total energy of the system considered

The first step to simplify the Schrödinger equation, for its solution, is to separate the nuclear and electron motions This procedure is called Born-Oppenheimer (BO) approximation [22] Within the BO approximation, the electrons are assumed to react instantaneously to any motion of the nuclei As a result of this assumption, the kinetic energy term of the nuclei can be omitted and the nuclear-nuclear repulsion is a constant for a given nuclear configuration The remaining terms

in the Hamiltonian constitute the so-called purely electronic Hamiltonian 𝐻̂el The wave function can thus be adequately approximated as a product of an electronic and

a nuclear wave function The electronic wave function only depends parametrically

on the nuclear coordinates Although the BO approximation does introduce errors to the calculations, these errors are usually small as compared to the errors of other approximations which are used to solve the Schrödinger equation The most importance consequence of the BO approximation is that the electronic energy plays the role of the potential energy in the equation of the nuclear motion

The potential energy surface, a multi-dimensional surface which describes how the electronic energy varies with respect to the nuclear coordinates, is a direct consequence of the BO approximation Analysis of a PES gives us much information about the chemical system of interest since ‘the PES is the most complete description

of all the conformers, isomers, and energetically accessible motions of a system.’ [23] These conformers/isomers correspond to the minima on a PES, the lowest-lying one of which being the global minimum, and the others being the local minima The

reactants and products of a reaction are connected by transition structures, which correspond to first-order saddle points on the surface The transition structures

obtained from the quantum chemical computations in the following chapters, are found by looking for the first-order saddle points along a reaction coordinate In a

simple, one-dimensional potential energy profile, which is usually constructed to probe the chemical reactions instead of the extremely complex PES, a TS corresponds

to the energy maximum of the potential energy curve along the reaction coordinate and the reactants/products correspond to the minima

2.2 Hartree-Fock theory

At this level of simplification, solution of the electronic Schrödinger equation

is still a formidable task for most chemical systems Further approximations must be used in order to make it more tractable, and thus different methods have been derived

Hartree-Fock (HF) theory is the simplest wave function-based method with the wave function Ψ0 being approximated as a single Slater determinant, written in terms of a set of spin-orbitals {𝑢i}, ΨSD = |𝑢1𝑢2… 𝑢n| [24] This expression of the

HF wave function satisfies the antisymmetric requirement of the wave functions of fermions (particles with half-integer spin like electrons) For atoms and molecules, these spin-orbitals are called atomic (AOs) and molecular spin-orbitals (MOs) Each spin-orbital is a product of a spatial orbitals and a spin function The variational principle tells us a way of finding the best HF wave function: the single Slater determinant which gives the lowest possible energy Since the HF wave function is composed of spin-orbitals, the problem of finding the best HF wave function becomes the problem of finding the best set of spin-orbitals which can minimize the energy of the system

The complicated many-body Schrödinger equation can be separated to form a set of single particle differential equations called Hartree-Fock equations, whose eigenfunctions and eigenvalues, for the case of molecules, are MOs and MOs energies, respectively For a system with n electrons and N nuclei, these equations have the forms:

N α

where 𝑓̂ is Fock operator, 𝑢i(𝑟⃗) are the orbitals, and 𝜖i are the orbital energies Note that these equations are written in atomic units The three terms in the Fock operator from left to right are kinetic energy, electron-nuclear potential energy, and the electron-electron potential energy which accounts for the potential energy of interaction between an electron and average field created by other electrons Because the HF wave function is a Slater determinant, the electron-electron potential energy contains extra terms beside the classical Coulomb 𝑗̂ terms These non-classical terms are called exchange 𝑘̂ operators

The HF equations must be solved iteratively because their solutions (orbitals) are needed to construct the Fock operator A HF calculation starts with an initial guess for a set of orbital coefficients These orbitals are used to construct the Fock operators and solving these HF equations results in a new set of orbitals This new set of orbitals

is then used in the next calculation step until the energies and orbital coefficients do not change from one step to the next This iteration procedure is called a self-consistent field procedure (SCF)

The orbitals can be written as linear combinations of a complete set of electron basis functions For MOs, these basis functions are usually chosen as atomic orbitals, the solutions of the Schrödinger equation for one-electron atoms The number of HF orbitals obtained after the calculations is equal to the number of basis functions that are used to expand the MOs The lowest-energy orbitals filled or

one-partially filled with electrons are called occupied orbitals The remaining empty orbitals are called virtual orbitals

The atomic orbitals can be represented by Slater- (STO) or Gaussian-type (GTO) orbitals that centred on the various atoms Although the former ones are more accurate, the integral calculations using them are much more time-consuming than using the latter type Therefore, a combination of several GTO orbitals can be used

to mimic an STO to reduce the computational cost The set of basis functions used to expand the molecular orbitals is called basis set, which is discussed in detail in section 2.6

If the basis set used is complete and thus infinite, we obtain the Hartree-Fock limit energy In practice the basis set employed is incomplete, and the HF energy obtained with this basis set is always higher than the HF limit energy Typically, the

HF theory can account for about 99% of the total energy of a system [25] The neglect

of the instantaneous interactions between electrons introduces an error to the HF calculation By definition, the difference between the exact non-relativistic energy and the HF limit energy is called electron correlation energy Note that the incompleteness of the basis set introduces an additional error to the calculation

Many efforts have been devoted to the evaluation of electron correlation in order to improve the electronic energy Correlated methods have thus been developed

to include the electron correlation, including the perturbation theory (for example, Møller-Plesset perturbation theory, MPn), configuration interaction (CI) theory, coupled-cluster (CC) theory, and multi-configurational self-consistent field theory (MCSCF)

2.3 Coupled-cluster methods

When the systems of interest have relatively small size (less than 10 heavy atoms), it is possible to use a more accurate method, coupled-cluster theory, to obtain better energetic data Similar to the configuration interaction method, in this CC method, the electron correlation is included by combining excited configurations with the reference wave function

The CI wave function is simply a linear combination of many configuration (state) functions (CSFs, Φ𝑖), each of which is a proper spin eigenfunction and can be composed of a single Slater determinant or a linear combination of several Slater determinants Of course, one can also write the CI wave function as a linear combination of Slater determinants In this case, the number of Slater determinants

will be larger than the number of CSFs The coefficients of the CSFs in the CI wave function are variationally determined The determinants used to construct the CSFs are formed from a set of spin-orbitals that obtained from a previous calculation (for example, a HF calculation) Depending on whether 1, 2, electrons are promoted from occupied to virtual spin-orbitals, the CSFs are categorized into several types called singly- or doubly-excited configurations, and so on

A difference between the CI and CC approaches is about the way of generating the wave function The CC wave function is obtained from applying an exponential cluster operator 𝑒𝑇̂ to the reference wave function Ψ0, which is usually but not necessarily, the HF wave function

in term of either the CSFs or the excited Slater determinants

Since the 𝑒𝑇̂ operator can be expanded into a Taylor-series, its effect on the

Ψ0 wave function is to express ψ as a linear combination of determinants, including

Ψ0 and excited determinants generated fromΨ0 In a CC calculation, the coefficients

or amplitudes of these determinants are optimized, whereas the orbitals are kept fixed

The CC calculations are classified based on the type of connected operators used to make the wave function, for example, CCSD includes 𝑇̂1 and 𝑇̂2, CCSDT includes 𝑇̂1, 𝑇̂2, and 𝑇̂3, etc When the triple contributions are included to the CCSD energy using the perturbation method, the resulting method is denoted as CCSD(T) [26] This method can give almost the same accuracy as the full CCSDT method with considerably smaller computational cost

2.4 Density-based methods

An alternative ab initio method is density functional theory, in which the total

energy is expressed in terms of the total electron density, rather than the wavefunction The two basic theorems of DFT were derived by Hohenberg and Kohn [27] The first Hohenberg-Kohn theorem showed that it is possible to express the ground state properties, particularly the total electronic energy E, of a system of interacting particles as a functional of the electron density The second Hohenberg-Kohn theorem allows us to introduce a variational principle for the total energy

of the kinetic energy and the electron-electron interaction energy results in the

exchange-correlation functional 𝐸[𝜌] This procedure maps the complicated many body problem of interacting electrons in the external potential of fixed nuclei onto a problem of non-interacting electrons moving in an effective external potential,

𝜐eff(𝑟⃗),

𝜐eff(𝑟⃗) = 𝜐ext(𝑟⃗) + 𝜐J(𝑟⃗) + 𝜐XC(𝑟⃗) (2.9)

where 𝜐ext(𝑟⃗) is the electron-nuclear potential or so-called external potential, 𝜐J(𝑟⃗)

is the classical Coulomb potential, and 𝜐XC(𝑟⃗) is the exchange-correlation potential

In other words, solutions for the complicated Schrödinger equation of the former problem can be obtained from a set of single-particle equations (Kohn-Sham equations), which are solvable when the form of 𝐸𝑋𝐶[𝜌] is known The canonical form of the Kohn-Sham equations is basically similar to that of the Hartree-Fock equations:

The LDA methods are based on uniform electron gas model in which the density is assumed as a slowly varying function The exchange-correlation energy can thus be expressed as a simple function of the density The GGA methods add the dependence on the gradient of the density into the exchange-correlation energy functional to allow the variation of the density In the hybrid functionals, a certain amount of the exact HF exchange is mixed into the pure DFT exchange 𝐸𝑋𝐶[𝜌] Because there exists no general procedure to improve the results of the DFT calculations as that of wave function based methods, suitable functional for system

of interest needs to be chosen carefully, usually by comparing with experiment or high accuracy computational methods

considered as the most accurate ab initio methods that are still applicable to

medium-sized organic systems [30]

Typical widely-used composite methods are the Gaussian-n (G2, G3, .) methods and the complete basis set (CBS) series (CBS-QB3, CBS-APNO, ) The latter set is developed to obtain the results at the basis set limit by extrapolating several single-point energies Another set of composite methods designed to obtain highly accurate thermodynamic predictions is the Weizmann-n series (W1, W2, W3, and recently W4) which incorporates corrections even past CCSD(T) correlation The high accuracy of these methods is thus accompanied by a significant computational cost

In the W1 method, optimized geometries are obtained using the density functional theory method B3LYP (Becke 3-parameter Lee-Yang-Parr) and the Dunning correlation consistent cc-pVTZ+1d basis set The characters +1d indicate the addition of high exponent d function to the set of basis functions of second-row elements The energies of the systems are calculated using the coupled cluster methods (CCSD and CCSD(T)) in conjunction with the aug-cc-pVQZ+2d1f and aug-

cc-pVTZ+2d1f basis sets These coupled cluster energies are then extrapolated to the expectation value at the complete basis set limit [31] The W1U method is a modification of the W1 method, in which unrestricted coupled cluster spin contamination corrected methods (UCCSD and UCCSD(T)) instead of restricted open-shell coupled cluster methods (ROCCSD and ROCCSD(T)) are used for open shell systems [20]

2.6 Basis sets

A basis set is a set of basis functions used to describe mathematically the orbitals within a system Using larger basis sets would impose fewer restrictions on the electrons, and thus would give a more accurately approximation of orbitals The choice of method and basis set affects significantly the accuracy of results and computation cost [32] Several types of basis sets are briefly discussed as follows

Minimal basis sets are ones in which a single basis function is used for each

orbital in the separated atoms that comprise a molecule Minimal basis sets use size atomic-type orbitals and typically give rough results that are insufficient for research-quality publication

fixed-A split basis set use one basis function for each core orbital, and several basis

functions for each valence orbital Split valence basis sets thus allow orbitals to change size, but not to change shape This type of basis sets is often used for larger molecules because of being less time consuming

To allow the variations of both size and shape of the orbitals, polarized basis functions with angular momentum beyond what is required for the ground state to description of each atom are added to the basis set In general, a basis function with

angular momentum l can be polarized by adding basis functions with angular momentum l+1 For example, d (l = 2) functions are added to the p (l = 1) basis functions of carbon and f (l = 3) functions are added to the d (l = 2) basis functions

of transition metals In some cases, the polarized p functions can also be added to the

s basis functions of hydrogen atoms Split-valence basis sets with added polarized

basis functions are called polarized basis sets

High angular momentum basis sets are larger basis sets which add multiple

polarization functions per atom to the triple zeta basis set Such basis sets are useful for describing the interactions between electrons in electron correlation methods

Similarly, the diffuse basis functions can be added to a basis set to describe more accurately the orbitals The diffuse functions are basis functions with very small exponents and decay slowly with distance from nucleus They are usually large-size versions of s- and p- type functions Basis sets with diffuse functions are important for systems where electrons are held far away from the nucleus: molecules with lone pairs, anions and other systems with significant negative charge, systems in their

excited states, systems with low ionization potentials, and so on

Correlation-consistent basis sets are developed by Dunning and coworkers for use in post-HF

correlated calculations [33] The correlation basis sets are built up by adding shell of basis functions to a core set of basis functions with the aim of converging smoothly

to the complete basis set limit For first- and second-row atoms, the basis sets are pVXZ, where X is cardinal number, which is the maximum angular momentum function in the basis set The aug- prefix can be used to add diffuse functions to the cc-pVXZ basis sets

cc-2.7 Standard statistical mechanic methods

Thermodynamic properties such as entropies and heat capacities could be calculated by using standard statistical mechanic methods Basically, thermodynamic properties of a molecule can be computed by adding up the contributions from the electronic, translational, external rotational, vibrational motions These contributions can be calculated from the corresponding partition function and its derivatives with respect to temperature The partition function can be calculated as follows:

where 𝐸𝑖 denotes the ith energy level, 𝑔𝑖 is the degeneracy of ith level, T is the

temperature, and kB is Boltzmann’s constant Overall partition function for N

molecules can be written as follows:

(2.17)

𝑞𝑒𝑙𝑒𝑐 = 𝑔×exp (−Δ𝐸

‡

Where m is mass, V is volume, s is the vibration mode which is 3N  6 for non linear molecules and 3N  5 for linear molecules,  is the vibrational frequency, I is the

moment of inertia, and 𝜎𝑟 is the rotational constant which can be obtained from the point group of the molecule Δ𝐸‡ is the energy difference between reactant and transition state

The contributions to other thermodynamic properties such as internal energy

U, heat capacity C and entropy S can be calculated in terms of the partition function

using the following equations:

2.8 Transition state theory

Transition state theory (TST) is the theory developed to explain the reaction rates of chemical reactions TST is based on some fundamental assumptions The first assumption is that the Born-Oppenheimer PES is valid, and one assumes the existence

of a hypersurface phase space that separates the reactant and product phase space

This hypersurface phase space is called a transititon state TST also assumes that

there is an equilibrium between transition state and reactants This transition state is considered as the kinetic bottleneck for the reaction Once a trajectory crosses the transition state, it leads directly to product without recrossing back to the reactant Another important consumption is that the energy in a molecule can, in a much shorter time than the time scale of any chemical reactions, be distributed into all possible internal degrees of freedom and each of these states have equal possibility

of occuring A molecule with a certain energy can thus be described on the basis of the number of quantum states accessible at that energy In all statistical rate theories the ratio between the number of quantum states at reactants and that at the transition state is the essential factor to calculate the rate coefficient

Canonical transition state theory (CTST) which can determine dependent rate constants is based on canonical energy distribution assumption In this assumption the energy populations of molecules are characterized by a thermal

temperature-equilibrium distribution According to CTST, the rate of a reaction can be expressed

in terms of the partition functions of the transition state and the reactants:

respectively Eb is the activation barrier at transition state

The ratio of the partition function of the transition state and that of the reactants quantifies the contribution of reactive quantum states over the reactant states, including contributions from translation, molecular rotation, and all internal degrees

of freedom The translational degree of freedom of the transition state is considered

as one that takes the system over a barrier and is written as the factor 𝑘B𝑇 ℎ⁄ The reaction coordidate can thus be excluded from the partition function of the transition state (indicated by symbol ‡)

Obtaining accurate rate predictions at any temperature requires an accurate prediction of the barrier height, and a correct description of the partition functions In addition, some other factors including tunneling effect, reactions in low pressure regime and symmetry factor are also needed to be taken into account Tunneling corrections can be very important in substitution reactions of light particles (H or D atoms) and should be thus incorporated into the rate equation by a factor 𝜅(𝑇) And for reactions that are not in the high pressure regime, insufficient number of collitions can cause CTST to be invalid In such cases, a falloff correction factor Γ(𝑃, 𝑇) needs

to be included to account for a noncanonical energy distribution as well as related recrossing effects

If multiple symmetrically equivalent reaction pathways are accessible, this can

be accounted for by a symmetry factor α as obtained from the ratio of the internal and external rotational symmetry numbers, σ, of reactants and transition state, the number

of optical isomers, m, and the electronic state degeneracy 𝑔:

The dynamics of a system including n wells and n products can be described

by the time-dependent, total-energy-resolved master equation as follows:

𝐸𝐼0

𝑛𝑤𝑒𝑙𝑙𝑠𝐼=1

(2.29)

In equation (2.28), the first term accounts for the gain of population 𝑁𝐼(𝐸) at energy level 𝐸 by collisional energy transfer from all other energy levels 𝐸′ with the collision probability 𝑃(𝐸, 𝐸′) and the collision frequency 𝜔𝐼 within isomer I; the

second term is for the loss by collisional energy transfer to other energy levels 𝐸′

within isomer I; the third term is for the loss due to the formation of reactant(s) 𝑅𝑚

to isomer I with the rate coefficient 𝑘(𝑅𝑚)𝐼(𝐸) (if they are connected); the fourth term

describes the reversible gain and loss from/to the other connected isomer J at energy

level 𝐸 with the microcanonical rate coefficients 𝑘𝐼𝐽(𝐸)and 𝑘𝐽𝐼(𝐸), respectively; the fifth term is for the gain from the reactant(s) at energy level 𝐸, 𝑁𝑅𝑚(𝐸); and the last one is for the irreversible loss to form product(s) with the corresponding microcanonical rate coefficients, 𝑘𝑃𝐼(𝐸), to the connected products The last term introduces an infinite sink approximation; thus it makes the system under consideration non-conservative In equation (2.29), the first and second terms describe the gain and the loss of population of the reactant(s) from and to the connected wells

Equations (2.28) and (2.29) form a set of ( nwells + 1) integro-differential equations for the population 𝑁1, 𝑁2, … , 𝑁𝑛𝑤𝑒𝑙𝑙𝑠 and 𝑁𝑅, that are very difficult to be solved analytically Alternatively, those equations can be discretized and solved

numerically, which was known as deterministic approach Once the matrix is

correctly diagonalized, the resulting eigenpairs can be used to derive time-resolved species profiles as well as the phenomenological rate coefficients of the system of interest using different approaches Upon diagonalization, the results of the ME equation, i.e., the concentration and their derivatives of the involved species as function of time, take the following form:

where 𝑥𝐼 and 𝑦𝑃 are mole fraction of the source species (wells and reactant) and the

sink species (products as in the nonconservative case); N is the dimension of the transition matrix; S and NP is the number of source species and products, respectively;

𝑐𝐼𝑖 and 𝑑𝑃𝑖 are expansion coefficients that depend on the initial distribution, the eigenvectors of the ME transition matrix and the microscopic rate coefficients from the source species to sink species (products)

Another approach is to obtain the species profiles (i.e., 𝑥𝐼 and 𝑦𝑃) directly by using the stochastic approach Equations (2.28) and (2.29) are solved by a finite number of trials (or particles/molecules at different conditions), 𝑁𝑡𝑟𝑖𝑎𝑙𝑠 In each trial, a particle

at a well I can be randomly initiated with an energy E (denoted as state (I,E)) Then, the time interval dt in which the particle stays at state (I,E) until the next “event” is

calculated by a random number 𝑟1:

𝑑𝑡 = −ln (𝑟1)

Where ∑(𝑘𝐽𝐼(𝐸), 𝜔𝐼) = ∑ 𝑘𝐽 𝐽𝐼(𝐸) + ∑ 𝜔𝐸′ 𝐼𝑃𝐼(𝐸′, 𝐸) ≡ ∑𝑛𝑖=1𝑎𝑖 is sum over all

possible rate coefficients from well I to J and collision frequency from E to E’ of well

I The next “event” j∈[1,n] which is either the collision or the reaction of the particle

is determined by a second random number 𝑟2:

∑ 𝑎𝑖

𝑗−1 𝑖

< 𝑟2∑(𝑘𝐼𝐽(𝐸), 𝜔𝐼) ≤ ∑ 𝑎𝑖

𝑗 𝑖=1

(2.32)

Where 𝑟1 and 𝑟2 have uniform distribution The progress is repeated until the particle

reach the product After a trial, we get a trajectory S(t) that tells us what species (i.e., reactants, intermediates and products) the particle is at time t, on which the profiles

𝑥𝐼 and 𝑥𝑃 can be derived by merging those S(t), and normalizing them with number

of trials Ntrials:

𝑁 𝑡𝑟𝑖𝑎𝑙𝑠

where 𝛿 is Kronecker function

2.10 Rate constant extraction

The rate constants were extracted using the Global-Minimum-Profile-Error (GMPE) procedure Let’s begin with the expression for the mole fractions of the source and sink species given in equation (2.30) The goal of the GMPE method is to find the macroscopic rate coefficient matrices that satisfy a set of differential rate equations:

where x and I x are given by the ME solution described above; therefore, it does not p

explicitly depend on the time-scale mixing The proposed GMPE method is applicable to any multiple-species and multiple-channel system at any condition as soon as the fitting is acceptable, which is controlled by the fitting tolerance parameters and can be quantified by the  value given by equation (2.34)

The key feature of the GMPE method is the introduction of the error term defined as follows

w t ; the ‘dot’ denotes ordinary derivative with respect to time t The weighting

function, w t( ), was chosen to selectively reflect time-domain of interest In general, the weighting functions can be any non-positive functions, locally integrable functions Currently, either a polynominal or t functions are implemented in the code The weighting function reflects the important time domain taken into account for fitting rate coefficients (e.g., limiting the chemical reaction timescale for such a purpose) It is observed that this is an important parameter to separate the chemical reaction time (from the lag time) for the case where the two time scales are mixing For the choice of w t ( )  t(the weighting function reflects the increase with time with positive ) implies that the short-time and long-time domains are considered less important Specifically, in the short-time domain, t is close to zero, thus the weighting function is close to zero; whereas at the long time limit, the concentration decay faster than the growing rate of the weighting function that make the product of the two is also close to zero

Taking the partial derivatives of equation (2.35) and then setting them to zero,

we have

1 0

which are numerically intergated if x is derived from stochastic solvating method, I

and are analytically integrated if x I derived from the eigenpairs u and IJ q IJ are then integrated as follows:

i j

IJ I J

t t

i j

IJ I J

t t

where U  (u IJ) and Q  (q IJ) are both S S square matrices The rate coefficient matrix is then given by

1,



whose elements, M IJ , denote the rate coefficient for reaction (between source species) JI

Analogously, the matrix for rate coefficients from the sources to the sinks can

be deduced by considering the error term

1, 2, , ; 1, 2, ,

N P

We also obtain the same results, with the lower Latin indices, i j, , running from 2 instead of 1

The GMPE was implemented in the Multi-Species Multi-Channel (MSMC) [35] code and then be used to derive rate coefficients of systems of interest