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
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:
CH3OO + NO → CH3O + NO2 (Rxn 1a)
An additional channel leading to the formation of CH2O and HONO was reported in several theoretical studies [8, 12, 13] as follows:
CH3OO + NO → CH2O + HONO (Rxn 2)
Lohr et al [8] found a transition structure (TS) for the CH2O elimination from methyl nitrate, which was found to occur at very low pressures (< 1 Torr) [13] Using the complete active space multiconfiguration self-consistent field (CASSCF) and complete active space with second-order perturbation theory (CASPT2) methods,
2 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 cm 3 ·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 cm 3 ·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 = k 1b/k 1a 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
3 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
THEORETICAL BACKGROUND
The time-independent Schrửdinger equation and the Born-Oppenheimer
Equivalent to the Newton’s law in classical mechanics, the fundamental equation of motion in quantum mechanics is the Schrửdinger wave equation postulated in 1926 [21], the time-independent form of which is:
The Hamiltonian operator 𝐻̂ of a system with N nuclei and n electrons can be expanded in the following form in atomic units:
5 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 Schrửdinger 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 𝐻̂ el 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 Schrửdinger 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
6 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.
Hartree-Fock theory
At this level of simplification, solution of the electronic Schrửdinger 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 Ψ 0 being approximated as a single Slater determinant, written in terms of a set of spin-orbitals {𝑢 i }, Ψ SD = |𝑢 1 𝑢 2 … 𝑢 n | [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 Schrửdinger 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:
(2.4) where 𝑓̂ is Fock operator, 𝑢 i (𝑟⃗) are the orbitals, and 𝜖 i 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 one- electron basis functions For MOs, these basis functions are usually chosen as atomic orbitals, the solutions of the Schrửdinger 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 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
8 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, Mứller-Plesset perturbation theory, MPn), configuration interaction (CI) theory, coupled-cluster (CC) theory, and multi-configurational self-consistent field theory (MCSCF).
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, Φ 𝑖 ), 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
9 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 𝑒 𝑇̂ to the reference wave function Ψ 0 , which is usually but not necessarily, the HF wave function
The cluster operator 𝑇̂ is a sum of a series of connected operators 𝑇̂ 1 , 𝑇̂ 2 , , and 𝑇̂ 𝑛 , which correspond to single, double, , and n-tuple excitations, respectively
In this case, we work with the excited Slater determinants instead of the CSFs
However, note that the CC wave function, similar to CI wave function, can be written in term of either the CSFs or the excited Slater determinants
Since the 𝑒 𝑇̂ operator can be expanded into a Taylor-series, its effect on the Ψ 0 wave function is to express ψ as a linear combination of determinants, including Ψ 0 and excited determinants generated fromΨ 0 In a CC calculation, the coefficients or amplitudes of these determinants are optimized, whereas the orbitals are kept fixed
The variational procedure to determine the coefficients of the determinants in the CC wave function is very difficult and thus a modification procedure is used
10 The CC calculations are classified based on the type of connected operators used to make the wave function, for example, CCSD includes 𝑇̂ 1 and 𝑇̂ 2 , CCSDT includes 𝑇̂ 1 , 𝑇̂ 2 , and 𝑇̂ 3 , 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.
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 calculation
The total energy functional has the following form:
𝐸[𝜌] = 𝑉 𝑛𝑒 [𝜌] + 𝑇[𝜌] + 𝑉 𝑒𝑒 [𝜌] (2.8) where 𝑉 𝑛𝑒 [𝜌] is the electron-nuclear interaction energy, 𝑇[𝜌] is the kinetic energy, and 𝑉 𝑒𝑒 [𝜌] is the electron-electron interaction energy
Kohn and Sham [28] showed how to ingeniously approximate the two big unknown terms, 𝑇[𝜌] and 𝑉 𝑒𝑒 [𝜌] functionals The 𝑉 𝑒𝑒 [𝜌] functional is separated into two parts, the Coulomb term 𝐽[𝜌] representing the well-known classical electron- electron repulsion and the unknown non-classical term 𝐸 𝑋𝐶 ′ [𝜌], which is much smaller than the Coulomb term, representing the non-classical exchange-correlation effect
Most of the kinetic energy 𝑇[𝜌] can be computed by introducing orbitals into the problem, leaving a small unknown part 𝑇 𝑐 [𝜌] Adding the two small unknown parts of the kinetic energy and the electron-electron interaction energy results in the
11 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, 𝜐 eff (𝑟⃗),
𝜐 eff (𝑟⃗) = 𝜐 ext (𝑟⃗) + 𝜐 J (𝑟⃗) + 𝜐 XC (𝑟⃗) (2.9) where 𝜐 ext (𝑟⃗) is the electron-nuclear potential or so-called external potential, 𝜐 J (𝑟⃗) is the classical Coulomb potential, and 𝜐 XC (𝑟⃗) is the exchange-correlation potential
In other words, solutions for the complicated Schrửdinger 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 𝐸 𝑋𝐶 [𝜌] is known The canonical form of the Kohn-Sham equations is basically similar to that of the Hartree-Fock equations:
DFT methods can be divided into different categories depending on how the 𝐸 𝑋𝐶 [𝜌] term is derived, such as local density approximation (LDA), generalized gradient approximation (GGA), and hybrid methods
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 𝐸 𝑋𝐶 [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
Composite methods
Although high level methods such as coupled-cluster methods with large basis sets can produce thermodynamic energies within chemical accuracy, which is known as an accuracy of 1 kcal·mol -1 , the computational cost in central processing unit (CPU) time scales extremely rapidly with the system size Other methods at lower levels of theory do not provide the reasonable accuracy One solution for this cost- accuracy compromise is to use composite methods which are multistep theoretical procedures commonly used to obtain highly accurate thermodynamic and kinetic data In the composite methods, different aspects such as geometry optimization, basis set convergence, inclusion of high-level correlation are individually optimized with the most cost-effective procedure The results of these separated procedures are subsequently combined to yield a level of accuracy close to that obtained from much more accurate-but-expensive calculations [29] The composite methods are thus 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-
13 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].
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 fixed- size atomic-type orbitals and typically give rough results that are insufficient for research-quality publication
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
14 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 cc- 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.
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:
15 where 𝐸 𝑖 denotes the i th energy level, 𝑔 𝑖 is the degeneracy of i th level, T is the temperature, and k B is Boltzmann’s constant Overall partition function for N molecules can be written as follows:
The partition function can be expressed as the product of the partition functions for translational (𝑄 𝑡𝑟𝑎𝑛𝑠 ), vibrational (𝑄 𝑣𝑖𝑏 ), rotational (𝑄 𝑟𝑜𝑡 ) and electronic energy (𝑄 𝑒𝑙𝑒𝑐 ) partition functions Each partition function can then be expressed as:
𝑘 𝐵 𝑇 ) (for transition states) (2.19) 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 𝜎 𝑟 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
16 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:
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 temperature- dependent rate constants is based on canonical energy distribution assumption In this assumption the energy populations of molecules are characterized by a thermal
17 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:
𝑘 B 𝑇) (2.23) where 𝑄 ‡ (𝑇) and 𝑄 reactants (𝑇) are the partition functions of the transition state and the reactants, respectively, ℎ and 𝑘 B are Planck’s constant and Boltzmann constant, respectively E b 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 𝑘 B 𝑇 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 symmetry number for molecular rotation is typically already incorporated in the partition function for molecular rotation; if so, then it should not be included in the calculation of α
The CTST rate expression is now given by:
An alternative expression of CTST rate is based on the heat of formation and entropy or Gibbs energy of activation as follows:
Where ∆𝐺 ‡ , ∆𝐻 ‡ , ∆𝑆 ‡ are the change in the Gibbs free energy, heat of formation and entropy between the transition structure and reactant, respectively.
Pressure-dependence analysis
The master equation (ME) framework were used to deal with temperature- and pressure-dependent chemical systems on multiple-compound multiple-channel potential energy surfaces Details about the methodology can be found elsewhere [34] A brief overview of ME framework deriving from previous work [35] is presented below
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:
In equation (2.28), the first term accounts for the gain of population 𝑁 𝐼 (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 𝜔 𝐼 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 𝑘 𝐼𝐽 (𝐸)and 𝑘 𝐽𝐼 (𝐸), respectively; the fifth term is for the gain from the reactant(s) at energy level 𝐸, 𝑁 𝑅𝑚 (𝐸); and the last one is for the irreversible loss to form product(s) with the corresponding microcanonical rate coefficients, 𝑘 𝑃𝐼 (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 ( n wells + 1) integro-differential equations for the population 𝑁 1 , 𝑁 2 , … , 𝑁 𝑛 𝑤𝑒𝑙𝑙𝑠 and 𝑁 𝑅 , 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:
𝑖=1 where 𝑥 𝐼 and 𝑦 𝑃 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 N P is the number of source species and products, respectively;
𝑐 𝐼𝑖 and 𝑑 𝑃𝑖 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., 𝑥 𝐼 and 𝑦 𝑃 ) 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), 𝑁 𝑡𝑟𝑖𝑎𝑙𝑠 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 𝑟 1 :
Where ∑(𝑘 𝐽𝐼 (𝐸), 𝜔 𝐼 ) = ∑ 𝑘𝐽 𝐽𝐼 (𝐸) + ∑ 𝜔 𝐸 ′ 𝐼 𝑃 𝐼 (𝐸 ′ , 𝐸) ≡ ∑ 𝑛 𝑖=1 𝑎 𝑖 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 𝑟 2 :
21 Where 𝑟 1 and 𝑟 2 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 𝑥 𝐼 and 𝑥 𝑃 can be derived by merging those S(t), and normalizing them with number of trials N trials:
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:
(2.34) where x I and x p are given by the ME solution described above; therefore, it does not 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
(2.35) which is the measure of square error, weighted by a non-negative function of time, ( ) 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
Within the eigenpairs ME framework, let
which are numerically intergated if x I is derived from stochastic solvating method, and are analytically integrated if x I derived from the eigenpairs u IJ and q IJ are then integrated as follows:
N i Ii Jj w i j i j u x x w t dt c e c e w t dt c c e w t dt c c L
Ii Jj w i j i j q x x w t dt c e c e w t dt c c e w t dt c c L
(2.39) where L w denotes the Laplace transform,
Thus, we have a set of linear equations for the rate matrix
24 where U ( u IJ ) and Q ( q IJ ) are both S S square matrices The rate coefficient matrix is then given by
M = UQ (2.41) whose elements, M IJ , denote the rate coefficient for reaction (between source species) JI
Analogously, the matrix for rate coefficients from the sources to the sinks can be deduced by considering the error term
Then the rate matrix describing reactions from sources to sinks is given by
K U Q (2.43) where U p is a N P S matrix whose elements are given by
The matrix K P contains the rate coefficients of reactions from source species to sinks species (products); explicitly the element k pI P is the rate coefficient of the reaction
For the conservative cases (i.e., no products/sinks present in the system), the first eigenvalue is identical to zero, 1 0 At the long time limit, the concentrations of source species do not vanish, x I x t I ( ) c I 1 Hence the error term is defined slightly differently
25 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
COMPUTATIONAL DETAILS
Electronic structure calculations
All calculations were performed using the Gaussian 09 package [36]
Electronic energies were obtained using the accurate composite W1U method [20]
In the W1U method, the molecular geometries are optimized using the density functional theory method B3LYP and the Dunning correlation consistent cc- pVTZ+1d basis set The energies are calculated using the unrestricted coupled cluster spin-contamination methods (CCSD and CCSD(T)) in conjunction with the aug-cc- pVDZ+2d and aug-cc-pVnZ+2d1f basis sets, where n = T, Q These coupled cluster (CC) energies are then extrapolated to the expectation value at the complete basis set limit The mean absolute deviations (MAD) and root-mean-square deviations (RMSD) between the total atomization energies, electron affinities, ionization potentials, and proton affinities calculated at the W1U level and the corresponding experimental data for the G2/97 test set are only 0.49 ± 0.27 and 0.65 ± 0.48 kcalˑmol -
1, respectively [20] Alternatively, the CCSD(T)/CBS//B3LYP/cc-pVTZ procedure was also used to obtain accurate energies where CBS denotes the extrapolation of cc- pVXZ results (X = D, T, Q) to the complete basis set limit and the exponential model,
A(X) = A() + B𝑒 −(𝑋−1) + C𝑒 −(𝑋−1) 2 , was used to fit the complete basis set limit [37]
All stable molecules as well as transition structures (TSs) were obtained for the lowest energy conformer of a given species Intrinsic reaction coordinate (IRC) [38, 39] calculations were performed at the same optimization method to confirm the minimum energy pathway (MEP) from the TSs to the corresponding reactant(s) and product(s).
Thermodynamic property and rate constant calculations
The molecular parameters of the reactants, products and TSs obtained from the electronic structure calculations were used to compute thermodynamic properties within the standard statistical mechanics framework The B3LYP/cc-pVTZ+1d harmonic frequencies were scaled by a factor of 0.985 prior to calculating zero-point
27 energy (ZPE) and thermodynamic properties [40] Low-frequency modes corresponding to internal rotations were treated as hindered rotors in which the corresponding hindrance potentials were computed at B3LYP/6-31G(d) level of theory via relaxed potential energy scan in the steps of 10 degrees of the dihedral angles corresponding to the rotation The averaged reduced moments of inertia required for the hindered rotation treatment were calculated with the proposal by East and Radom [41], based on the original work of Kilpatrick and Pitzer [42], described in the work of Mai et al [43] The Schrửdinger equation could be solved numerically by using the wave function of the free rotor as basic functions The obtained energy levels are used to yield the partition function and contribution to the thermodynamic function f H, entropy S, and heat capacity C p The atomization method was employed to calculate the standard heat of formation while statistical mechanic methods were used to calculate the other parameters with the W1U information
The high-pressure rate constants for elementary reactions were calculated using canonical transition state theory with the tunneling correction based on asymmetric Eckart potentials [44] The high-pressure kinetic data for the barrierless radical recombination were adopted from the work of Barker et al [7] Pressure- and temperature-dependent rate constants for the multi-well system were calculated within the RRKM/ME framework The estimated Lennard-Jones (LJ) collision diameters σ of 4.3 Å and well depths ε/k B of 484 K were used for the adducts and isomers [7] LJ parameters for the bath gases of He and N2 are σ = 2.551 Å, ε/k B 10.0 K and σ = 3.798 Å, ε/k B = 71.4 K, respectively [45] We adopted the averaged energy transferred in a deactivating collision = 100 cm -1 and 250 cm -1 for the colliders He and N2, respectively [46], which are the bath gases used in the corresponding experiments All thermodynamic and rate constant calculations were calculated using the Multi-Species Multi-Channels (MSMC) code [35] with the aid of the MSMC-GUI (Graphical User Interface) [47]
MSMC is a parallel C/C++ ab initio-based code which is divided into five main modules (cf Figure 3.1) Some selected information from electronic structure
28 calculations (e.g., Gaussian output) was used to construct MSMC input in the first module, Preprocessing In the second module, Precalculation, thermodynamic properties of all species and other necessary information were computed and provided for the third module, Matrix Setup and Diagonalization The aim of third module is to set up and diagonalize the transition matrix to get its eigenpairs Related information was mentioned in section 2.9 The Rate Coefficient Extraction and Analysis module is to analyze the results and extract rate coefficients Different output files are created to assist the analysis of the results, each with different purposes (e.g., thermodynamic/kinetic analyses and detailed subkinetic mechanisms for flame/reactor modeling) The last module is the Postprocessing to examine and analyze the results on the fly [35]
Figure 3.1 Flowchart of the Multi-Species Multi-Channel (MSMC) program [35]
RESULTS AND DISCUSSION
OO + NO reaction
Potential energy surface
An extensive effort was made to construct detailed potential energy surface of the reaction CH3OO + NO using the accurate composite W1U method Note that hindered internal rotation treatment was rigorously included to improve the harmonic oscilator approximation It is worth mentioning that in this study two new product channels, namely, dehydration (P3) and association (P4) were found; thus, their role was examined in the kinetic analysis
The complete PES at the W1U level was shown in Figure 4.4 The relative energies of selected reaction channels, obtained by different methods, as well as the corresponding literature data were provided in Table 4.1 A good agreement between different sets of data was observed Lesar et al [12] performed a theoretical study using G3 and CCSD(T) methods The single point CCSD(T) calculations were carried out using MP2/6-311++G(d,p) optimized geometries The mean absolute deviations between our numbers with these two levels are 2.3 and 0.7 kcalˑmol -1 , respectively The W1U values are close to those recently reported by Launder et al
[48] using the focal point approach (FPA) [49-51] with the MAD of 0.6 kcal·mol -1 It is encouraging that the W1U and CCSD(T)/CBS//B3LYP numbers are very close to each other with the maximum difference of 0.3 kcal·mol -1 and the MAD of 0.1 kcal·mol -1 Therefore, confidence was gained on the W1U data which were then used in the detailed kinetic analysis
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)
Table 4.1 Relative zero-point corrected energies (at 0 K in kcal·mol -1 )for selected species involved in the reaction
Energy at 0 K (kcal·mol -1 ) Species W1U a CCSD(T)/CBS/
CH3OONOcp (IM1) -23.8 -23.7 -24.9 -21.0 -24.6 CH3OONOtp (IM2) -22.4 -22.3 -23.5 -19.6 -23.2 CH3ONO2 (IM3) -53.3 -53.2 -53.0 -47.5 -53.8 CH3O + NO2 (P1) -11.8 -12.0 -11.6 -11.7 -12.4 HCHO + HONO (P2) -70.1 -70.2 -70.4 -70.0 -70.4
Maximum difference 0.3 1.3 5.8 1.3 a This work b Computed using the focal point approach [48] c From the work of Lesar et al [12] d Mean absolute deviation
Initially, CH3OO and NO can form either cis-perp CH3OONO (IM1) or trans- perp CH3OONO (IM2), two distinct conformers of the methyl peroxynitrite adduct, characterized by the dihedral angle OONO That IM1 is slightly more stable than
IM2 (the bond energy of -23.8 and -22.4 kcal·mol -1 , respectively) agrees well with literature data [12, 48] Once formed, the two conformers can either isomerize to each other with a barrier of 11.6 kcal·mol -1 below the entrance channel or dissociate to CH3O + NO2 (P1) via O–N bond cleavage with no TS [13, 48] The isomerization barrier from IM1 to IM2 was found to be 12.2 kcal·mol -1 at W1U level of theory, which is close to those reported by Lesar et al [12] (12.2 kcal·mol -1 at G3 level) and
Launder et al [48] (13.0 kcal·mol -1 ) The IM1P1 and IM2P1 dissociation pathways are endoergic by 12.0 and 10.6 kcal·mol -1 , respectively
The two initially-formed adducts, IM1 and IM2, can isomerize to CH3ONO2
(IM3) or HOCH2NO2 (IM4), respectively Lesar et al [12] reported a TS for the
IM2IM3 isomerization Zhang et al [11] also assumed that only IM2 can isomerize to IM3 in the kinetic study of R-ONO2 formation in the reaction of RO2 + NO (R = H, CH3) In this study, we confirmed, by IRC calculations, that only TS-1,3
36 connects IM1 to IM3, while TS-2,4 is the TS between IM2 and IM4 This conclusion is consistent with the suggestion about the existence of a TS between IM3 and IM1 by Lohr et al [8] and Zhao et al [18] Although IM4, lying about 80.9 kcal·mol -1 below the entrance channel, is more stable than IM3, the formation of IM4 via TS-
2,4 has a much higher barrier compared to the formation of IM3 via TS-1,3 (15.1 vs
-2.5 kcal·mol -1 relative to the entrance channel) Consequently, the former channel is expected not to play a role compared to the other low-lying channels The dissociation of CH3ONO2 to CH3O + NO2 products (P1) is endoergic by 41.5 kcal·mol -1
The HCHO + HONO products (P2) can be formed by either IM3 or IM4 dissociations through five-member ring TSs The W1U barrier height for IM3P2 dissociation pathway is 44.1 kcal·mol -1 , which agrees with the CCSD(T)/CBS//B3LYP number (43.7 kcal·mol -1 ) and the literature values (43.3 and 42.8 kcal·mol -1 ) [12] (see Table 4.2) The dissociation of IM4 to products P2 is endoergic by 10.8 kcal·mol -1 , and the corresponding barrier height of this pathway is 23.5 kcalˑmol -1 Alternatively, IM4 can proceed further through isomerization channel, in which one H atom transfers from hydroxylmethyl group to O atom of nitrate group via a four-member ring TS TS-4,5, to form HOCH=N(=O)OH (IM5) with the barrier of 63.2 kcal·mol -1 Intermediate IM5 can isomerize to form
O=CHN(OH)2 (IM6) via a five-member ring TS TS-5,6 with the barrier of 8.8 kcal·mol -1 IM6 can either dehydrate to O=CHN=O + H2O (P3) or isomerize to form O=CHNH(=O)OH (IM7) which subsequently yield O=CHONHOH (P4) The two channels energetically comparable with the energy of 81.8 and 82.3 kcalˑmol -1 , below the entrance channel, respectively
Table 4.2 Comparison of the calculated reaction barrier heights (at 0 K in kcal·mol -1 ) with the available literature data
Reaction barrier height, V ‡ (kcal·mol -1 ) W1U a
(IM3 P2) 44.1 43.7 40.9 43.3 42.8 a This work b From the work of Lesar et al [12] c The exponential model, A(X) = A() + B𝑒 −(𝑋−1) + C𝑒 −(𝑋−1) 2 , was used to fit the complete basis set limit [37]; X is the cardinal number of the Dunning’s correlation consistent basis set cc-pVXZ, where X = 2, 3, 4 for n = D, T, Q, respectively [52-54].
Thermodynamic properties
Calculated thermodynamic properties of methylperoxy radical CH3OO including heat of formation, electron affinity (EA), ionization energy (IE), proton affinity (PA), and O–H bond dissociation energy of HOOCH3 were presented and compared with available literature in Table 4.3 It can be seen that the W1U results are consistent with available theoretical and experimental data Using atomization method, our W1U enthalpy of formation at 298 K (2.74 kcal·mol -1 ) agrees well with the experimental values of 2.7 ± 0.8 [55] and 2.92 ± 0.22 [56] kcal·mol -1 and the calculated value of 2.37 kcal·mol -1 by Wang and Bozzelli [57].Also, our predicted electron affinity, ionization energy and O–H bond dissociation energy values also match well with literature values Such good agreements on different thermodynamic parameters have help to gain more confidence on the W1U method for the title system CH3OO + NO Thermodynamic properties for selected important species were presented in Table 4.4 while those for all related species can be found in Appendix Table A3 The MAD and RMSD between our calculated numbers with the data from Active Thermochemical Tables (ATcT) [58], Webbook National Institute of
38 Standards and Technology (NIST) [59] and Burcat’s thermodynamic database [60] are available in Appendix Table A4 In comparison with the ATcT [58] data, the MAD and RMSD for f H(298 K) are 0.35 and 0.48 kcal·mol -1 , respectively In comparison with the NIST [59] data, the maximum difference is about 1 kcal·mol -1 Meanwhile, the maximum differences between the W1U values and the Burcat’s thermodynamic data [60] for f H(298 K), S(298 K)and C p(T) are 0.82 kcal·mol -1 , 2.53 cal·mol -1 ·K -1 and 0.67 cal·mol -1 ·K -1 , respectively Table 4.5 presents the calculated energetic data of some selected dissociation reactions in comparison with literature data The W1U values in this work are in good agreement with the
CASPT2//CASSCF values reported by Arenas et al [13] The maximum difference of the dissociation enthalpy between W1U and CASPT2//CASSCF is 1.8 kcal·mol -1 The good agreements indicate that our thermodynamic results can be used for development of the kinetics and reaction mechanisms to understand the chemistry and model these processes
Table 4.3 Calculated thermochemical parameters for methylperoxy radical
(CH3OO) Numbers are in kcal·mol -1 , otherwise, they will be noted
87.8 ± 0.1 [62] a Values collected from Updated Active Thermochemical Tables (ATcT) values based on ver 1.118 of the Thermochemical Network, http://atct.anl.gov (access date:
Table 4.4 Calculated thermodynamic properties of selected species involved in the reaction and available experimental/theoretical data (ATcT = Active Thermochemical Tables a ,NIST = Webbook NIST b , and Burcat = Burcat’s thermodynamic database c ) Units: kcal·mol -1 for f H(298 K)and cal·mol -1 ·K -1 for S and C p
Burcat -57.79 45.13 8.03 8.19 8.42 8.68 9.26 9.87 11.31 a Values collected from Updated Active Thermochemical Tables (ATcT) values based on ver 1.118 of the Thermochemical Network, http://atct.anl.gov (access date: April 2017) b Values collected from NIST chemistry webbook, http://webbook.nist.gov/chemistry/ (access date: Dec 2016) c Values collected from Burcat’s online database, http://garfield.chem.elte.hu/Burcat/burcat.html (access date: Dec 2016) d Data calculated at W1U level of theory e f H(298 K) calculated by atomization method
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
CH 3 OONOcp → CH 3 OO + NO
CH 3 OONOtp → CH 3 OO + NO
Kinetic calculations/analysis
The specificities of the mechanism and kinetic characteristics of the reaction CH3OO + NO were elucidated by analyzing the calculated time-resolved species profiles The calculated profiles at two different temperatures (298 and 1500 K) and 760 Torr pressure were shown in Figure 4.5 It can be seen that the mechanism was initiated by forming two excited methyl peroxynitrite intermediates, IM1 and IM2
The mole fraction of IM1 is greater than that of IM2 by a factor of approximately 20
This is not surprising since cis-perp conformer (IM1) is energetically more stable than trans-perp conformer (IM2) by 1.4 kcal·mol -1 and the barrier of the isomerization between two conformers is 12.2 kcal·mol -1 The excited methyl peroxynitrite intermediates could either proceed further to form products, isomerize to form methyl nitrate CH3ONO2 (IM3) or HOCH2NO2 (IM4), or undergo collisional stabilization The mole fractions of two conformers increase rapidly until reach equilibrium at nearly 10 -14 second and then start to decrease at 10 -3 second (see Figure 4.5a) At room temperature, CH3O + NO2 (P1) and HCHO + HONO (P2) can be formed by CH3ONO2 (IM3) dissociations The IM3P2 dissociation requires a high
42 barrier of 44.1 kcal·mol -1 while that of IM3P1 has no TS The mole fraction of IM3 increases gradually until reach equilibrium at the value of 7.65 10 -14 for a short period and then increase to the asymptotic value of 2.68 10 -9 Even though the mole fraction of IM3 is quite small compared to that of P1, our predicted result indicates that a highly excited IM3 can be collisionally deactivated to form stable methyl nitrate thus accounting for the detection of methyl nitrate in the experimental work of Butkovskaya et al [16] The primary product is CH3O + NO2 (P1), whose mole fraction keep increasing, accounting for 99% of product formation eventually
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) See Figure 4.4 for species notations
At higher temperature (namely 1500 K, cf Figure 4.5b), the concentration of
IM1 and IM2 are comparable, and the other thermodynamically favorable product channels such as dehydration (P3) and association (P4) start to play a noticeable role even minor Because of the shift in the distribution at the higher temperature according to the Maxwell–Boltzmann distribution [68], a greater fraction of molecules has enough kinetic energy to overcome the barrier and hence there is a greater chance to produce more the other products Thus the increase of temperature has larger effect on the high barrier channels, namely P3 and P4 Additionally, at higher temperature, IM1 molecules can easily to isomerize to IM2, thus there is no significant difference in mole fractions of IM1 and IM2 After the equilibrium is
43 reached, the mole fraction of IM3 is 2.08 10 -14 and then decreases sharply, indicating that at higher temperature collisional stabilization of IM3 is inefficient, compared to the lower temperature
In order to investigate the pressure effect, the time-resolved species profiles were representatively examined at two different pressures (100 and 760 Torr) at 298 K, plotted in Figure 4.6 Overall, no significant pressure effect on P1 and P2 formation channels were observed There is a growth in the mole fraction of IM3 as pressure increases from 100 to 760 Torr Since the collisional stabilization forming
IM3 becomes progressively more competitive Our conclusion agrees with the experimental study of Butkovskaya et al [16] carried out at the temperature of 298 K and pressure range of 50–500 Torr By using a turbulent flow reactor coupled with a chemical ionization mass spectrometer, these authors showed a positive dependence of IM3 formation on the pressure
Figure 4.6 Time-resolved species profiles for the reaction CH3OO + NO at 298 K and two different pressures (100 and 760 Torr) See Figure 4.4 for species notations
The high-pressure rate constants in the temperature range of 300–1500 K for all elementary reactions were calculated using the canonical transition state theory with the tunneling correction using the asymmetric Eckart potential The calculated rate constants in the modified Arrhenius format were tabulated in Table 4.6 The total
44 rate constants (CH3OO + NO products) at different temperatures and pressures were plotted and compared with literature data in Figure 4.7 In general, the overall rate constants slightly increase with decreasing temperature The agreement is relatively good in the considered range (T = 193–298 K & P = 4–200 Torr) Our calculated results are within a factor of two when compared to different experimental numbers Masaki et al [4] used time-resolved measurements of the radical ion signals and Xing et al [10] used a laser flash photolysis coupled with time-resolved negative ionization mass spectrometry technique to investigate the CH3OO + NO reaction at low pressure (< 5 Torr) Our rate constant at 298 K was predicted to be 16.1 10 -12 cm 3 ·molecule -1 ·s -1 , which is in good agreement with experiment results of (11.2 ± 1.4) 10 -12 by Masaki et al [4] and (9.9 ± 2.1) 10 -12 cm 3 ·molecule -1 ·s -1 by Xing et al [10] Turbulent flow technique and mass spectrometry were used by both Scholtens et al [5] and Bacak et al [9] to determine the rate constant at higher pressures of 100 and 200 Torr It is clear that our predicted rate constants are more consistent with the numbers investigated by Scholtens et al [5] than Bacak et al [9]
Additionally, the rate constants calculated at below 243 K are obviously in much better agreement with literature results than those predicted at above 243 K It should be noted that the high-pressure rate constants for barrierless channels suggested by Barker et al [7] were used for the pressure-dependence analysis To investigate the sensitivity of the calculated pressure-dependent rate constant on the assigned rate constants, the uncertainty of ± 50% of the assigned numbers were used for such an analysis With such uncertainty, our calculated rate constants satisfactorily cover all experiment data including the observed negative temperature dependence (cf Figure 4.7)
Table 4.6 Calculated high-pressure rate constants a for selected reaction pathways of the title reaction
10.33 33.04 a Rate constants were presented by k(T) = A T n exp(-E a /RT), valid for temperature range 300–1500 K Units: s -1 for first-order reaction and cm 3 ·molecule -1 ·s -1 for second-order reaction
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 [9], Masaki et al [4], Scholtens et al [5], Xing et al [10] The experimental studies were carried out over the pressure range 4–200 Torr and the temperature range 193–298 K
4.4.3 Pressure- and temperature- dependent analysis
There are few reports investigating the temperature dependence of the rate coefficient for the reaction CH3OO + NO It can be seen from Figure 4.7 that there is an evidently negative temperature dependence of the calculated rate constant, which is consistent with the conclusions of previous studies at room temperature [5, 9] The calculated rate constant decreases roughly by 50% as temperature increases from 193 K to 298 K
Figure 4.8 presented the predicted values for elementary rate constants for the CH3OO + NO reaction in the extend temperature range 300–1500 K and pressure range 76–76000 Torr The rate constant for the formation of CH3O + NO2 (k P1 ) are significantly higher than those of other products more than four order of magnitude, indicating that the main product of this reaction is CH3O + NO2 at considered
47 conditions As can be seen from Figure 4.8, at low pressure (e.g., 76 Torr) the values of rate constants k P1 , k IM3 considerably increase with the temperature, which reflects the positive temperature dependence At higher pressures in Figure 4.8b and 4.8c the collisional deactivation forming IM3 becomes progressively more competitive The value of rate constant k IM3 is greater than k P2 at 76000 Torr and above 650 K
Moreover, it turns out that there is a negative temperature dependence of k IM3 at the pressure of 76000 Torr The temperature dependence of P2 channel is rather complicated For example, at a fixed pressure (e.g., 76, 7600 and 76000 Torr) there is a minimum value as temperature changes and this value slightly increase with pressure
We also carried out an analysis of the effect of pressure on the rate constant of CH3OO + NO reaction Butkovskaya et al [16] investigated experimentally the formation of CH3ONO2 and the temperature and pressure dependence of CH3ONO2 formation in the reaction of CH3OO + NO They proposed that CH3ONO2 formation shows a small dependence on pressure At 298 K, there was a growth in the CH3ONO2 yield with increasing pressure from 0.33 ± 0.16% to 0.80 ± 0.54% over the pressure range 50–500 Torr In this study, we further extended our study to the pressure- dependent behavior over the temperature range of 300–1500 K It can be seen that while the pressure has an unnoticeable effect on k P1 , the rate constant k IM3 displays a positive pressure dependence (see Figure 4.8d, 4.8e) At low temperatures, the pressure-dependent behavior of k IM3 is consistent with the work of Butkovskaya et al [16] However, at higher temperature, particularly at 1500 K, the rate constant k IM3 remains stable over the pressure period studied Other thermodynamically unfavorable product channel (e.g., P2) starts to play a minor role at high temperatures (see Figure 4.8d and 4.8f)
CONCLUSIONS AND OUTLOOK
In this thesis, the accurate composite W1U method was used to intensively explore the CH3OO + NO potential energy surface on which its time-resolved temperature- and pressure- dependent mechanism was elucidated within the RRKM/ME framework
The reaction was found to proceed through the two conformers of the methyl peroxynitrite adduct, IM1 and IM2 Four products P1 (CH3O + NO2), P2 (HCHO + HONO), P3 (CHONO + H2O), and P4 (HCOONHOH) can be formed from the methyl peroxynitrite adduct The potential energy surface showed that the product P1 is the most kinetically favorable product and P2 is the second most feasible one This is the first time that P3 (CHONO + H2O) and P4 (HCOONHOH) product channels for the reaction CH3OO + NO were reported in a theoretical study However, two newly-found products P3 and P4 are minor products because their formations involve higher barrier consumed and complicated processes
Thermodynamic properties were carried out based on the molecular parameters derived from the electronic structure calculations The good agreements between our calculated data and literature ones shows that W1U method can provide a reasonable estimation of electronic structures and thermodynamic results With no adjustment, the calculated kinetic data were found in good agreement with experimental values, and more insights related to the pressure- and temperature- dependence of rate constants were revealed The CH3O + NO2 channel was found dominate in the considered conditions (T = 298–1500 K & P = 7.6–76000 Torr)
Moreover, the study suggested the RRKM/ME rate models can be used as a powerful post-facto and predictive tool for kineticists The detailed kinetic sub-mechanism, including both thermodynamic data in NASA polynominal format and kinetic data in modified Arrhenius format can be used for further detailed modeling and simulation of NOx conversion
50 To improve our understanding of the chemistry of combustion processes, other studies will be carried out in the near future: one first applies reaction rate parameters and thermodynamic data as input for combustion models using CHEMKIN program package, and the impact of the uncertainties of input parameters on simulation output will be investigated by using sensitivity analysis Other factors such as the effect of roaming transition states on this reaction and potential energy surface associated with the triplet electronic state also can be studied in the future work
1 H.T Nguyen, T.V.-T Mai, L.K Huynh, Detailed kinetic mechanism for CH3OO + NO reaction – An ab initio study, Comput Theor Chem 1113,
2 H.T Nguyen, L.K Huynh, T.N Truong, Migration and desorption of hydrogen atom and molecule on/from graphene, Carbon 121, (2017), 248-256
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Table A1 Conventional names, short notations and 2-D structures for all species
Cis-perp methyl peroxynitrite CH3OONO
Trans-perp methyl peroxynitrite CH3OONO
(hydroxymethylidene)azinic acid HOCH=N(=O)OH
Table A2 Harmonic vibrational frequencies of all species including reactants, intermediates, products and transition states calculated at B3LYP/cc-pVTZ+1d level of theory Electronic energies calculated at W1U level of theory
Table A3 Calculated thermodynamic properties of species involved in the reaction and available experimental data (ATcT Active Thermochemical Tables a , NIST = Webbook NIST b and Burcat = Burcat’s thermodynamic database c ) Units: kcal·mol -1 for
f H(298 K)and cal·mol -1 ·K -1 for S and C p
(cis-perp) This work d -0.02 76.99 19.92 22.87 25.46 27.67 31.10 33.59 37.31 CH3OONO (trans- perp) This work d 1.46 76.61 20.56 23.57 26.14 28.29 31.58 33.91 37.33
Burcat -18.81 60.72 11.10 12.41 13.52 14.43 15.69 16.58 17.88 HOCH2NO2 This work d -57.55 74.99 18.21 21.98 25.22 27.82 31.52 33.97 37.38 HOCH=N(=O)OH This work d -48.29 72.30 20.19 24.03 27.28 29.99 33.92 36.34 39.06 O=CHN(OH)2 This work d -53.28 73.65 20.83 24.34 27.09 29.25 32.37 34.55 37.87
O=CHNH(=O)OH This work d -35.59 73.10 20.00 23.83 27.30 30.21 34.25 36.56 39.06 O=CHONHOH This work d -58.99 71.99 18.75 22.44 25.55 28.07 31.77 34.34 38.20 CH3O
TS-7,P4 This work d -26.29 68.51 16.79 20.71 23.92 26.46 30.14 32.67 36.44 a Values collected from Updated Active Thermochemical Tables (ATcT) values based on ver 1.118 of the Thermochemical Network, http://atct.anl.gov (access date: April 2017) b Values collected from NIST chemistry webbook, http://webbook.nist.gov/chemistry/ (access date: Dec 2016) c Values collected from Burcat’s online database, http://garfield.chem.elte.hu/Burcat/burcat.html (access date: Dec 2016) d Data calculated at W1U level of theory e f H(298 K) calculated by atomization method
Table A4 Comparison between mean absolute deviation (MAD) and root-mean-square deviation (RMSD) of W1U/ MSMC a , ATcT b , NIST c , Burcat d Units: kcal·mol -1 for f H(298 K)and cal·mol -1 ·K -1 for S and C p
RMSD 0.50 1.14 0.22 0.24 0.28 0.34 0.34 0.37 0.29 a This work; b Values collected from Updated Active Thermochemical Tables (ATcT) values based on ver 1.118 of the Thermochemical Network, http://atct.anl.gov (access date: April 2017) c Values collected from NIST chemistry webbook, http://webbook.nist.gov/chemistry/ (access date: Dec 2016) d Values collected from Burcat’s online database, http://garfield.chem.elte.hu/Burcat/burcat.html (access date: Dec 2016)
Table A5 A comparison of rate constants for the reaction CH3OO + NO
Rate constants, 10 -12 cm 3 ·molecule -1 ·s -1 This work Expt’l