QUANTUM CHEMICAL STUDIES AND KINETICS OF GAS REACTIONS Except where reference is made to the work of others, the work described in this disertation is my own or was done in collaboration with my advisory commite. This disertation does not include proprietary or clasified information. Hasan Sayin Certificate of Approval: Thomas E. Albrecht-Schmit Michael L. McKe, Chair Asociate Profesor Profesor Chemistry and Biochemistry Chemistry and Biochemistry Thomas R. Webb Rik Blumenthal Asociate Profesor Asociate Profesor Chemistry and Biochemistry Chemistry and Biochemistry Joe F. Pitman Interim Dean Graduate School QUANTUM CHEMICAL STUDIES AND KINETICS OF GAS REACTIONS Hasan Sayin A Disertation Submited to the Graduate Faculty of Auburn University in Partial Fulfilment of the Requirements for the Degre of Doctor of Philosophy Auburn, Alabama December 15, 2006 ii QUANTUM CHEMICAL STUDIES AND KINETICS OF GAS REACTIONS Hasan Sayin Permision is granted to Auburn University to make copies of this disertation at its discretion, upon request of individuals of institutions and at their expense. The author reserves al publication rights. Hasan Sayin Date of Graduation iv VITA Hasan Sayin, the first child of Batal Sayin (father) and Esma Sayin (mother), was born on June 07, 1978, in Malatya, Turkey. He graduated from Bogazici (Bosphorus) University in Istanbul, Turkey with a Bachelor of Science degre in Chemistry on July 4, 2002. He entered the Ph.D. program in the Department of Chemistry, Auburn University in August, 2002. He joined Dr. Michael McKe's research group and studied quantum mechanical calculations and kinetics of gas reactions. v DISERTATION ABSTRACT QUANTUM CHEMICAL STUDIES AND KINETICS OF GAS REACTIONS Hasan Sayin Doctor of Philosophy, December 15, 2006 (B.S Bogazici (Bosphorous) University, 2002) 138 Typed Pages Directed by Michael L. McKe Potential energy surfaces and reaction mechanisms were calculated using various computational methods such as density-functional and wave-function methods. High- level computational methods were used to obtain acurate rate constants. Theoretical background and computational methods are introduced in Chapter 1. Chapter 2 and Chapter 3 show potential energy surface and kinetic calculations of important atmospheric reactions. Chapter 4 covers the theoreticaly chalenging molecule F 2 NOF which has not yet been synthesized experimentaly. The potential energy surface and the rate constants for reaction of XO (X=Cl, Br, I) with dimethyl sulfide (DMS) have been computed at high levels of theory in Chapter 2. vi Natural bond orbital (NBO) analysis of XO-DMS and the branching ratios of the each pathways are computed. In Chapter 3 the reaction of NO with ClO has been studied theoreticaly using density-functional and wave-function methods (B3LYP and CSD(T). Variational transition-state theory was used to calculate the rate constant for disappearance of reactants (k dis ) and for formation of products (k obs ) in the range of 200-1000K at the high- presure limit. The mechanism of disociation of F 2 NOF has been studied using various computational methods in Chapter 4. Rate constant calculations have been performed to understand the product formation. The calculated results showed that the formation of F 3 NO is favored thermodynamicaly but not favored kineticaly. vii ACKNOWLEDGMENTS I would like to acknowledge Prof. Michael L. McKe for seting a high standard of excelence for me and for giving me wonderful opportunities to do research in his group. He has been an invaluable source of knowledge and inspiration during my graduate life in Auburn. Dr. Nida McKe deserves special mention for her help in my graduate life in Auburn. I would like to thank my commite members, Prof. Thomas E. Albrecht-Schmit, Prof. Thomas R. Webb and Prof. Rik Blumental for their contributions towards my disertation. I am also grateful to my friends, Eren Orhun, Denem Orhun and their parents, Prof. Emrah Orhun, Dr. Deniz Orhun for their encouragement and help in Auburn. Finaly, if there were not heroic eforts of my father Batal Sayin and my mother Esma Sayin and the patience of my sisters G?l Sayin and Demet Sayin, al of whom supported and encouraged me to take my Ph.D., I certainly could never have acomplished anything. vii Style Manual or Journal manual used: Journal of the American Chemical Society Computer software used: Mac-MS word 2001, CS ChemDraw Std. 9.0 for Mac, KaleidaGraph 3.6 ix TABLE OF CONTENTS LIST OF FIGURES ............................................... xi LIST OF TABLES ................................................ xv CHAPTER 1 GENERAL INTRODUCTION ............................ 1 1.1 Schr?dinger Equation ..................................... 3 1.2 The Born-Oppenheimer Approximation ....................... 4 1.3 Hartre-Fock Theory ...................................... 6 1.4 Basis set ............................................... 7 1.5 Restricted and Unrestricted Hartre-Fock ...................... 11 1.6 Electron Correlation Methods ............................... 12 1.7 Kinetics ................................................ 18 1.7.1 Bimolecular Reactions ................................... 19 1.7.1.a Collision theory ....................................... 19 1.7.1.b Transition State Theory ................................. 20 1.7.1.c Variational Transition State Theory ........................ 22 1.8 Unimolecular Reactions ................................... 22 1.8.1 RKM Theory ......................................... 25 1.8.2 The High Presure Limit ................................... 28 x 1.8.3 The Low-Presure Limit .................................. 28 1.8.4 Adiabatic Rotations ..................................... 30 1.9 References ............................................. 31 CHAPTER 2 COMPUTATIONAL STUDY OF THE REACTIONS BETWEN XO (X=Cl, Br, I) DIMETHYL SULFIDE ....................... 34 2.1 Introduction ............................................ 34 2.2 Method ................................................ 37 2.3 Results and discussion ..................................... 42 2.3a ClO+DMS ............................................. 42 2.3b BrO + DMS ........................................... 47 2.3c IO + DMS ............................................. 50 2.3d Bonding in XO-DMS Adducts ............................. 54 2.4 Conclusions ............................................ 57 2.5 References ............................................. 58 CHAPTER 3 THEORETICAL STUDY OF THE MECHANISM OF NO 2 PRODUCTION FROM NO + ClO ......................... 63 3.1 Introduction ............................................ 63 3.2 Computational Method .................................... 65 3.3 Results and Discusion ............................................. 71 3.4 Rate calculations ......................................... 77 3.5 Conclusion ............................................. 84 xi 3.6 References ............................................. 85 CHAPTER 4 THE DISOCIATION MECHANISM OF STABLE INTERMEDIATE: PERFLUOROHYDROXYLAMINE ....................... 90 4.1 Introduction ............................................ 90 4.2 Computational Method .................................... 93 4.3 Results and Discussions .................................... 96 4.4 Stability of F 2 NO......................................... 107 4.5 Rate Constant Calculations ................................. 110 4.6 Conclusion ............................................. 116 4.7 References ............................................. 117 xii LIST OF FIGURES CHAPTER 2 Figure 1. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p) level. Bond lengths are in ? and angles are in degres. The optimized structure of Cl-abst-TS is at the MP2/6-31G(d) level. ................................... 45 Figure 2. Schematic diagram of the potential energy surface computed at the G3B3 (298K) level for the reaction of ClO with DMS. Relative energies are given in kcal/mol at 298 K. ........................... 46 Figure 3. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p) level. Bond lengths are in ? and angles are in degres. ........................................... 48 Figure 4. Schematic diagram of the potential energy surface computed at the G3B3(MP2) (298K) level for the reaction of BrO with DMS. Relative energies are given in kcal/mol at 298 K. .................... 49 Figure 5. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p)/ECP level. Bond lengths are in ? and angles are in degres. ........................................ 51 Figure 6. Schematic diagram of the potential energy surface computed at the xii G2B3(MP2) (298K) level for the reaction of IO with DMS. Relative energies are given in kcal/mol at 298 K. .................... 52 Figure 7. Interaction diagram of the singly occupied molecular orbital (SOMO) of XO (X=Cl, Br, I) interacting with the highest occupied molecular orbital (HOMO) of DMS. ....................... 55 CHAPTER 3 Figure 1. Optimized geometry of cis-, trans-ONOCl and ClNO 2 isomers. Bond lengths are in ? and angles are in degres. Data in the first row and third row are calculated at the CSD(T)/c-pVDZ and B3LYP/6-311+G(d) level, respectively. (a) Calculated values at the CSD(T)/TZ2P and B3LYP/TZ2P level, respectively are taken from Ref. 19. (b) and (c) are experimental values, are taken from Ref. 42 and Ref. 52, respectively. .......................... 70 Figure 2. Optimized geometric parameters of stationary points at the CSD(T)/c-pVDZ level with B3LYP/6-311+G(d) values in parentheses. Bond lengths are in ? and angles are in degres. The geometric parameters for ON-OCl-ts-c and ON-OCl-ts-t are at the RCSD(T)/c-pVDZ level with values at the UCSD(T)/c-pVDZ level given in brackets. ................................. 74 Figure 3. Schematic diagram of the potential energy surface for the NO + ClO system computed at the CSD(T)/c-pVTZ/CSD(T)/c-pVDZ level. Relative energies xiv are given in kcal/mol at 298K. ........................... 71 Figure 4. Ilustration of cis and trans chlorine addition to NO 2 to form (a) cis- ONOCl and (b) trans-ONOCl. Cis addition can be rationalized by an orbital mixing mechanism. Trans addition has a higher activation barier and involves an electron promotion mechanism. ......... 76 Figure 5. Calculated and experimental rate constants for the ClO + NO reaction. Al references are experimental rate data except Ref. 16 which is computational. The computational rate data are for the bimolecular rate constant at the high-presure limit. The thin lines for the cis and trans rate constants are added together to give k dis 80 Figure 6. Plot of vibrational frequencies (cm -1 ) along the IRC for trans ? cis isomerization with the reaction projected out. The level of theory is B3LYP/6-311+G(d). ................................... 81 Figure 7. Comparison of ln(k) versus 1/T for transition state theory (TST) and variational transition state theory (CVT). The ChemRate results are RKM. ............................................. 83 CHAPTER 4 Figure 1. Optimized geometry of cis-, trans-F 2 NOF and F 3 NO isomers. Bond lengths are in ? and angles are in degres. For each isomer, methods are shown with the data. The last row of F 3 NO isomers are experimental values. (a) Frost, D. C.; Hering, F. G; Mitchel, K. A. R; Stenhouse, I. R. J. Am. Chem. Soc. 1971, 93, 1596. ..... 95 xv Figure 2. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d) level. Values in parentheses are at the CSD/6-31+G(d) level and values in brackets are at the CASCF/6-311+G(d) level. Bond lengths are in ? and angles are in degres. ........................................... 98 Figure 3. Schematic diagram of the potential energy surface for the disociation of F 2 NOF system computed at the CSD(T)/c- pVQZ/B3LYP/6-311+G(d) level. Relative enthalpies are given in kcal/mol at 298K. ..................................... 103 Figure 4. Interaction diagram comparing (a) the zwiterionic transition state (Add-F-N-ts) and (b) the biradical transition state (F-F 2 NO-ts). ... 103 Figure 5. Schematic diagram of the potential energy surface ?H(0K) for the disociation of F 2 NO at the CSD(T)/c-pVQZ/CSD/c-pVDZ level. Values in parentheses are at the CSD(T)/c-pVQZ/CSD(T)/c-pVDZ level. ............... 108 Figure 6. Optimized geometric parameters of stationary points at the CSD/c-pVDZ level. The parameters in parentheses are optimized at the CSD(T)/c-pVDZ level. .......................... 109 Figure 7. Calculated rate constants involving cis-F 2 NOF. The computational rate data are for the unimolecular rate constant at the high-presure limit. ............................................... 112 Figure 8. Calculated rate constants involving trans-F 2 NOF. ............. 114 xvi LIST OF TABLES CHAPTER 2 Table 1. Experimental and theoretical rate constant (cm 3 ?s -1 ) for XO + DMS (X=Cl, Br, I) at 298K .................................. 36 Table 2. Spin-orbit corrections a (SOC) in kcal?mol -1 .................. 39 Table 3. Relative energies (kcal?mol -1 ) at the DFT and ab initio levels for various species involved in the XO + DMS Reactions (X=Cl, Br, I) 41 Table 4. Calculated rate constants (cm 3 ?s -1 ) and branching Ratios of XO + DMS (X=Cl, Br, I) at 298K for Oxygen-Atom Transfer (OAT) and Hydrogen Abstraction Pathways. .......................... 53 Table 5. Experimental Binding Enthalpies (kcal?mol -1 ) of X?X - , Calculated Binding Enthalpies (kcal?mol -1 ) of XO?OX - , and PseudoIE (eV) of X - and XO - ........................................ 56 Table 6. NPA Charges, Total Atomic Spin Densities, and ?-bond Polarization of the ? so bond in the XO-DMS complex (X=Cl, Br, I) ......................................... 57 xvii CHAPTER 3 Table 1. Relative Energies (kcal/mol) at the B3LYP/6-311+G(d), G3B3, CSD(T)/c-pVTZ/CSD(T)/c-pVDZ Levels for Various Species Involved in the NO + ClO Reaction. ................. 67 Table 2. Harmonic frequencies of trans-ONOCl, cis-ONOCl and ClNO 2 in cm -1 . ............................................... 73 CHAPTER 4 Table 1. Harmonic frequencies of cis-F 2 NOF, trans-F 2 NOF and F 3 NO in cm -1 . ............................................... 96 Table 2. Relative Enthalpies (kcal/mol) for Various Species Involved in the Disociation of F 2 NOF. .............................. 97 Table 3. Spin Density, Natural Population Analysis (NPA) and Geometry are calculated at the (U)B3LYP/6-311+G(d) level. ............ 104 Table 4. Enthalpies and fre energies of Fluorine loss reaction at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) level. .............. 105 Table 5. Enthalpies of the various types of F 2 NO species optimized at the CSD/cpVDZ level. ............................... 106 Table 6. Spin Densities and Mulliken Charges for c-F-FNO-c s and F-FNO-c 1 ............................................ 109 Table 7. Rate constant with temperature dependence at high-presure limit for formation of F 3 NO and disociation of complex to radicals. ..... 113 xvii The research presented herein has resulted in the following publications: Sayin, H.; McKe, M. L. Computational Study of the Reactions betwen XO (X=Cl, Br, I) and Dimethyl Sulfide, J. Phys. Chem. A 2004, 108, 7613. Sayin, H.; McKe, M. L. Theoretical Study of the mechanism of NO 2 production from NO + ClO, J. Phys. Chem. A 2005, 109, 4736. Sayin, H.; McKe, M. L. Disociation Mechanism of a Stable Intermediate: Perfluorohyrodxylamine, J. Phys. Chem. A, 2006, 110, 10880. 1 CHAPTER 1 GENERAL INTRODUCTION Chemicals structures and reactions are simulated numericaly by computational algorithms that are based on the fundamental laws of physics. Chemists can study a chemical event by running calculations on computers rather than by doing reactions and synthesizing compounds experimentaly. Unstable intermediates and transition states can be modeled by computational chemistry, which can provide information about molecules and reactions that is impossible to obtain through observation alone. There are two broad areas in computational chemistry, molecular mechanics 1 and electronic structure theory 1 . They involve the same basic types of calculations such as computing the energy of a particular molecular structure, performing geometry optimizations, and computing the vibrational frequencies of molecules. Molecular mechanics predicts the structures and properties of molecules by using clasical physics but does not treat the electrons in a molecular system. Electronic efects are included in force fields through parametrization. Molecular mechanics computations are inexpensive computationaly and are used for very large systems containing thousands of atoms. However this technique has many limitations. Since it neglect electrons, it cannot treat chemical problems where electronic efects are dominant. For example, it cannot describe proceses that involve bond formation or bond breaking. 2 If we are interested in describing the electron distribution in detail, there is no way other than quantum mechanics. Electronic structure methods use quantum mechanics rather than clasical physics. Electrons are very light particles and cannot be described correctly by clasical mechanics. Quantum mechanics implies that the energy and the other properties of a molecule can be obtained by solving the Schr?dinger equation: 1,2,3 ?? = E? (1) The exact solutions for the Schr?dinger equation are not computationaly practical. Therefore various mathematical approximations are applied to solve the Schr?dinger equation. There are thre major clases of electronic structure methods which are semi- empirical methods, 4,5 ab initio methods 6 and density functional methods. 7 Semi-empirical methods such as AM1, 8 MINDO/3 8,9 and PM3 10,1 use parameters derived from experimental data to simplify the computation. They solve an approximate form of the Schr?dinger equation. Diferent semi-empirical methods are clasified by their parameter sets. Semi-empirical calculations are relatively inexpensive compared to ab initio and provide reasonable descriptions of molecular systems and fairly acurate in predictions of energies and structures for many systems. Ab initio methods use no experimental parameters in their computations. Their computations depend on the laws of quantum mechanics. Ab initio computations provide high quality predictions for many systems. A third clas of electronic structure methods used widely is caled density functional methods (DFT). DFT methods are similar to ab initio methods. DFT is les 3 expensive than ab initio methods. Since they include the efects of electron corelation, they can give the benefits of some more expensive ab initio methods at a cheaper cost. The field of computational fluid dynamics, proces simulation and design, combustion, and atmospheric chemistry are just a few of the areas that need acurate rate constants for chemical reactions. Predicting rate constants is in fact a major goal of computational chemistry. Calculations of rate constants require the acuracy of the dynamical theory and the eficiency in obtaining acurate potential-energy information. Direct dynamics calculations provide a practical approach to the calculation of chemical reaction rates and achieve a greater understanding of dynamical bottlenecks, tunneling mechanisms, and kinetic isotope efects in chemical reactions. Direct dynamics approach is based on the output (energies, gradients, and hesians) of electronic structure calculations. Their weaknes is a fairly high cost in terms of computer time because so many electronic structure calculations are involved. 1.1 Schr?dinger Equation The Austrian physicist Erwin Schr?dinger proposed an equation to find the wavefunction of any system in 1926. The Schr?dinger equation for a particle of mas m moving in one dimension with energy E is given in eq 2. ! " h 2 2m d 2 # dx 2 +V(x)#=E# (2) 4 V(x) is the potential energy of the particle and depends on the position x; ? ? =h/2? is the modification of Planck's constant (h) and ? is the wave function of the system with respect to time. The energy and many other properties of the particle can be obtained by solving the Schr?dinger equation for ?. For many real-world problems the energy distribution does not change with time t, and it is useful to determine how the stationary states change with position x (independent of the time t). For every time-independent Hamiltonian H, there exist a set of quantum states, ? n , known as energy eigenstates and corresponding real numbers E n satisfying the eigenvalue equation in eq 3. ?? n (x)=E n ? n (x) (3) Such a state has a definite total energy, whose value E n is the eigenvalue of the state vector with the Hamiltonian. This eigenvalue equation is refered as the time- independent Schr?dinger equation. Equation 3 is a non-relativistic description of the system, which is not valid when the velocities of particles approach the speed of light. 1.2 The Born-Openheimer Aproximation As nuclei are much heavier than electrons, their velocities are much smaler. Therefore the Schr?dinger equation can be separated into two parts; one part describes the electronic wave function for a fixed nuclear geometry, and another part describes the nuclear wave function, where the energy from the electronic wave function plays the role of a potential energy. This separation is caled the Born-Oppenheimer (BO) 5 approximation. 1,2 In another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei in BO approximation. The full Hamiltonian for the molecular system can be writen as given in eq 4. ? =T elec + T nucl + V nucl-elec + V elec + V nucl (4) T and V terms are kinetic and potential energy terms, respectively. The Born- Oppenheimer approximation alows solving two parts of the problem independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term from the nuclei as given in eq 5. ? elec =T elec + V nucl-elec + V elec + V nucl (5) This Hamiltonian is used in the Schr?dinger equation to describe the motion of electrons in the field of fixed nuclei shown in eq 6. ? elec ? elec = E eff ? elec (6) Solving this equation for the electronic wavefunction wil produce the efective nuclear potential function E eff . It depends on the nuclear coordinates and describes the potential energy surface for the system. E eff is also used as the efective potential for the nuclear Hamiltonian, shown in eq 7. 6 H nucl = T nucl + E eff (7) This Hamiltonian is used in the Schr?dinger equation for nuclear motion, to describe the vibrational, rotational, and translational states of the nuclei. Solving the nuclear Schr?dinger equation approximately is necesary for predicting the vibrational spectra of molecules. 1.3 Hartre-Fock Theory Hartre-Fock theory 1,2 is one of the fundamental concepts of electronic structure theory. It depends on molecular orbital (MO) theory, that uses one-electron wave function or orbitals to construct the full wave function. Molecular orbitals are the product of Hartre-Fock theory, and Hartre-Fock is not an exact theory: it is an approximation to the electronic Schr?dinger equation. The approximation is that we pretend that each electron fels only the average Coulomb repulsion of al the other electrons. This approximation makes Hartre-Fock theory much simpler than the real problem, which is an N-body problem. Unfortunately, in many cases this approximation is rather serious and can give poor answers. It can be corrected by explicitly acounting for electron correlation by density functional theory (DFT), many-body perturbation theory (MBPT), 12 configuration interaction (CI), 13,14 and other means. It is important to remember that these orbitals are mathematical constructions which approximate reality. Only for hydrogen atom (or other one-electron systems, like He + ) are orbitals exact eigenfunctions of the full electronic Hamiltonian. As long as we are content to consider molecules near their equilibrium geometry, Hartre-Fock theory 7 often provides a good starting point for more elaborate theoretical methods which are beter approximations to the electronic Schr?dinger equation. 1.4 Basis set The approximation involves expresing the molecular orbitals as linear combination of a pre-defined set of one-electron functions known as basis functions. 15 These basis functions are usualy centered on the atomic nuclei and so bear some resemblance to atomic orbitals. Larger basis sets more acurately approximate the orbitals by imposing fewer restrictions on the locations of the electrons in space. An individual molecular orbital is defined in eq 8, ! " i =c ?i # ?1 N $ (8) where the coeficients c ?i are known as the molecular orbital expansion coeficients. ? ? refers to an arbitrary function in the same way ? i refers to an arbitrary molecular orbital. Ab initio methods try to derive information by solving the Schr?dinger equation without fiting parameters to experimental data. Actualy, ab initio methods also make use of experimental data, but in a somewhat more subtle fashion. Many diferent approximate methods exist for solving the Schr?dinger equation, and the one to use for a specific problem is usualy chosen by comparing the performance against known experimental data. Therefore experimental data guides selection of the computational model, rather than directly entering the computational procedure. 8 Basis sets 2 are one of the approximations inherent in esentialy al ab initio methods. Expanding an unknown function, such as a molecular orbital, in a set of known functions is not an approximation, if the basis set is complete. However, a complete basis set means that an infinite number of functions must be used, which is impossible in actual calculations. The smaler the basis, the poorer the representation. There are two types of basis functions commonly used in quantum mechanics calculations. These are Slater-type orbitals (STO) 16 and Gaussian-type orbitals (GTO). 1,2 STOs are not appropriate for numerical computations of multi-centered integrals while solving the Schr?dinger equation due to high cost in computer time. Therefore their practical use in quantum-mechanical calculations is now limited. Eventhough most quantum mechanics programs use GTOs as basis functions, GTOs have dificulty in describing the proper behavior near the nucleus. Neverthles, GTOs have the important advantage that evaluating a GTO integral while solving the Schr?dinger equation takes much les computer time than a STO integral evaluation. Therefore, GTOs are prefered and are generaly used in computational calculations. STOs and GTOs functional forms are given in eq 9 and eq 10, respectively. ! " #,n,lm (r,$,%)=NY l,m ($,%)r n&1 e &#r 9 ! " #,n,lm (r,$,%)=Nx l x y l y z l z e &#r 2 10 After deciding the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smalest number of functions possible is a minimum basis set. For hydrogen and helium this means a single s-function. 9 For the first row in the periodic table it means two s-functions (1s and 2s) and one set of p-functions (2p x , 2p y and 2p z ). For the second-row elements, thre s-functions (1s, 2s and 3s) and two sets of p-functions (2p and 3p) are used. The next improvement in the basis sets is a doubling of al basis functions, producing a Double Zeta (DZ) type basis set. 1,2 A DZ basis set employs two s-function for hydrogen (1s and 1s'), four s-functions (1s, 1s', 2s and 2s') and two sets of p-functions for first-row elements, and six s-functions and four sets of p-functions for second-row elements. For instance, 3-21G basis have two sets of functions in the valence region. The number of basis function for 3-21G and 6-31G basis sets are the same, but the 6-31G basis set has more GTOs than 3-21G. Therefore a 6-31G basis set describes molecular properties beter than a 3-21G basis set. However we can use 3-21G for very large molecules for which 6-31G is too expensive. Doubling the number of basis functions alows a much beter description of electron distribution in chemical bonding. The chemical bonding occurs betwen valence orbitals. For example doubling the 1s- functions in carbon alows for a beter description of the 1s-electrons. However, the 1s- orbital is independent of the chemical environment. A variation of DZ type basis only doubles the number of valence orbitals, producing split valence basis sets. In actual calculations a doubling of the core orbitals would rarely be considered, and the term DZ basis is also used for split valence basis sets or sometimes VDZ, for valence double zeta. 1 The next step in basis set size is a Triple Zeta (TZ). 1 This basis set contains thre time as many functions as the minimum basis set, i.e. six s-functions and thre p- functions for the first row elements. Some of the core orbitals may again be saved by only spliting the valence, producing a triple split valence basis set. For example, a 6- 10 311G basis set uses thre sets of basis functions for each valence atomic orbital. The names Quadruple Zeta (QZ) and Quintuple Zeta (5Z) for the next-level basis sets are also used. 1 In most cases higher angular-momentum functions are also important; these are denoted polarization functions. 1,2 For example, the C-H bond in HCN is primarily described by the hydrogen s-orbitals and the carbon s- and p z -orbitals. It is clear that the electron distribution along the bond wil be diferent than that perpendicular to the bond. If only s-functions are present on the hydrogen, this can not be described. However, if a set of p- orbitals is added to the hydrogen, the p z component can be used for improving the description of the C-H bond. The p-orbital introduces a polarization of the s-orbitals. Similarly d- orbitals can be used for polarizing p-orbitals, f-orbitals for polarizing d- orbitals etc. 6-31G(d) basis set is a VDZ polarized set which adds six d-type functions to the 6-31G basis set on each atom other than hydrogen. The 6-31G(d,p) basis set adds thre p-type functions to the 6-31G(d) basis set on each hydrogen. The basis sets can be also improved by adding difuse functions that are large-size versions of s- and p-type functions. 1,2 They alow orbitals to occupy a larger region of space. Basis sets with difuse functions are important for systems where electrons are relatively far 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. The 6-31+G(d) basis set is formed from the 6-31G(d) basis set by adding four difuse functions (s, p x , p y , p z ) on each non-hydrogen atom. The 6-31++G(d) set also includes one difuse s-type function on each hydrogen atom. 11 Basis sets for atoms beyond the third row of the periodic table are handled diferently. For these very large nuclei, electrons near the nucleus are treated in an approximate way, via efective core potentials (ECPs). 17 This treatment includes some relativistic efects, which are important in these atoms. The ECPs and asociated basis sets of Hay-Wadt, 18 Stevens and co-workers, 19 and Stuttgart-Dresden ECP 20 are widely used and implemented in many computational chemistry packages. 1.5 Restricted and Unrestricted Hartre-Fock If the system has an even number of electrons and a singlet type of wave function (a closed-shel system), such wave functions are known as Restricted Hartre-Fock (RHF). 17 Open-shel systems may also be described by the restricted type wave functions where the part of the doubly occupied orbitals is forced to be the same: this is known as Restricted Open-shel Hartre-Fock (ROHF). 17 In other words, it uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons For open-shel systems, an unrestricted method, 17 capable of treating unpaired electrons is needed. For this case, the alpha and beta electrons are in diferent orbitals, resulting in two sets of molecular orbital expansion coeficients. The two sets of coeficients result in two sets of Fock matrices and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper disociation to separate atoms, correct delocalized orbitals for resonant systems, and other atributes characteristic of open-shel systems. However eigenfunctions are not pure spin states, but contain some 12 amount of spin contamination from higher states (for example, doublets are contaminated to some degre by states corresponding to quartets and higher states). 1.6 Electron Correlation Methods The Harte-Fock method generates solutions to the Sch?dinger equation where the real electron-electron interection is replaced by an average interaction. With sufficiently large basis sets, the HF wave function is able to acount for 99% of the total energy, but the remaining 1% is often very important for describing chemical phenomena. The diference in energy betwen the HF and the lowest possible energy in a given basis set is caled the Electron Correlation (EC) energy. 10 There are four main methods for calculating electron correlation: Configuration Interaction (CI), Many-Body Perturbation Theory (MBPT), Coupled-Cluster (C) 21 and Density Functional Theory (DFT). The HF method determines the best one-determinant trial wave function within the given basis set. Therefore it is clear that in order to improve on HF results, the starting point must be trial wave function which contains more than one determinant. As the HF solution usualy gives 99% of the correct answer, electron correlation methods normaly use the HF wave function as a starting point for improvements. Configuration Interaction (CI) methods begin by noting that the exact wave function ? can not be expresed as a single determinant, as HF theory asumes. The additional determinants beyond the HF are constructed by interchanging one or more occupied orbitals within the Hartre-Fock determinant with a virtual (unoccupied) orbital. These can be denoted acording to how many occupied HF MOs have been replaced by 13 virtual MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These determinants are often refered to as Singles (S), Doubles (D), Triples (T), Quadruples (Q) etc. If al possible determinants (full CI) in a given basis set are included, al the electron correlation (in the given basis) can be recovered. For an infinite basis set the Schr?dinger equation is then solved exactly. Methods which include electron correlation are two-dimensionals, the larger the one-electron expansion (basis set size) and the larger the many-electron expansion (number of determinants), the beter are the results. The multi-configuration self-consistent field (MCSCF) 3 can be considered as a CI where not only the coeficients in front of the determinants are optimized, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure. (If the "multi-configuration" is only one, it is simply HF) Increasing the number of configurations in an MCSCF wil recover more and more of the dynamical corelation (correlating the motion of the electrons), until at the full CI limit, the correlation treatment is exact. MCSCF methods are mainly used for generating a qualitatively correct wave function. The major problem with MCSCF methods is selecting the necesary configurations to include for the property of interest. One of the most popular approaches is the Complete Active Space Self-Consistent Field (CASCF) method (also caled Full Optimized Reaction Space, FORS). 2 The selection of configurations is done by partitioning the MOs into active and inactive spaces. The active MOs wil typicaly be some of the highest occupied and some of the lowest unoccupied MOs from a HF 14 calculation. The inactive MOs either have 2 or 0 electrons, i.e. always doubly occupied or empty. Within the active MOs, a full CI is performed, and al the proper symmetry- adapted configurations are included in the MCSCF optimization. The active space must be decided manualy, by considering the problem and the computational expense. A common notation is [n,m]-CASCF, indicating that n electrons are distributed in al possible ways in m orbitals. Another approach to electron correlation is M?ller-Pleset perturbation theory. 1,2 Qualitatively, M?ller-Pleset perturbation theory adds higher excitations to Hartre-Fock theory as a non-iterative correction, using the techniques from the mathematical physics known as many-body perturbation theory. Perturbation theory is based upon dividing the Hamiltonian (eq 11) into two parts: ? = ? 0 + ?? ' (11) ? 0 is a reference Hamilton operator and ?' is a perturbation Hamilton operator. ? is variable parameter determining the strength of the perturbation. The zero-order wave function is the HF determinant, and the zero-order energy is just a sum of MO energies. Since the first-order energy is exactly the HF energy, electron correlation starts at order 2 with the choice of H 0 (se eq 12 and eq 13). MP0 = E(MP0)= ! "i i=1 N # (12) MP1=MP0 + E(MP1)=E(HF) (13) 15 The lowest-energy solution to the unperturbed problem is the HF wave function, additional higher-energy solutions are excited Slater determinants, analogously to the CI method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. Therefore the expansion of the many-electron wave function is truncated. Wel-known truncated MP methods are MP2 (truncation after 2nd- order), MP3, MP4 and so forth. MP2 typicaly acounts for 80-90% of the correlation energy, and it is the most economical method for including electron correlation. High order of MP methods such as MP3 and MP4 are available in case MP2 is inadequate. However, MP4 is more used compared to MP3 because MP3 calculations are known to provide litle improvement over MP2 results. 1 Perturbation methods add al types of corrections (S,D,T,Q, etc.) to the reference wave function to a given order (2,3,4,etc.). The idea in Coupled Cluster (C) methods is to include al corrections of a given type to infinite order. C methods include the efects of single and double excitation, efectively adding higher-order excitation than MP4. C calculations are similar to CI methods in that the C wavefunction is expresed as an exponential of excitations operating on a single Slater determinant, which solves the size- consistency problem of CI methods. The cluster operator is a sum of excitations of singles and doubles, etc. These excitations are comonly truncated at double excitations to give Coupled Cluster Singles and Doubles (CSD). 23 Like QCISD(T) 24 , triple excitations are included perturbatively to CSD, which is caled CSD(T) 24 . Coupled Cluster (CSD) and quadratic (QCISD) methods with triple excitations (CSD(T) and QCISD(T) often gives similar quality in terms of acuracy of energies and geometries. 16 Standard coupled-cluster theory is based on a single-determinant reference wave function. Coupled cluster is somewhat more tolerant of a poor reference wave function than MP methods due to the summation of contributions to infinite order. Since the singly excited determinants alow the MOs to relax in order to describe the multi- reference character in the wave function, the magnitude of the singles amplitude is an indicator of how good the HF single determinant is as a reference. The quality of CSD wave function is evaluated by the T 1 -diagnostic. 25 Specificaly, if T 1 < 0.02, the CSD(T) method is expected to give results close to the full CI limit for the given basis set. If T 1 is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should be employed. The foundation for Density Functional Theory (DFT) is the proof by Hohenberg and Kohn 1,2,26 that the ground-state electronic energy is determined completely by the electron density ?. Following on the work of Kohn and Sham, the approximate functionals employed by current DFT methods partition the electronic energy into several terms where E T is the kinetic energy term, E V is the potential energy of the nuclear- electron atraction and of the repulsion betwen pairs of nuclei, E J is the electron-electron repulsion term and E XC is the exchange-correlation term and includes the remaining part of the electron-electron interactions (eq 14). E XC includes the exchange energy arising from the antisymmetry of the quantum mechanical wavefunction and dynamic correlation in the motions of the individual electrons. E= E T + E V + E J + E XC (14) 17 E XC is usualy divided into exchange and correlation parts, which actualy correspond to same-spin and mixed-spin interactions, respectively (eq 15). E XC (?) = E X (?) + E C (?) (15) Pure DFT methods are defined by pairing an exchange functional with a correlation functional. For example, the wel-known BLYP functional pairs Becke's gradient-corrected exchange functional with the gradient-corrected correlation functional of Le, Yang and Par. 27,28 Self-consistent Kohn-Sham DFT calculations are performed in an iterative manner that is similar to an SCF computation. This similarity to the methodology of Hartre-Fock theory was pointed out by Kohn and Sham. E XC is formulated by including the mixture of Hartre-Fock and DFT exchange along with DFT correlation by Becke (eq 16). ! E hybrid XC =c HF E HF X +c DFT E DFT XC (16) where the c's are constants. B3LYP 29 functionals are currently the most widely used functionals for molecular calculations because they give good equilibrium geometries, vibrational frequencies, and acurate reaction energies. Even though DFT methods include electron correlation, they take almost the same computer resources as HF calculations which neglect electron correlation. It is also known that B3LYP results are equal to MP2 in terms of geometries and relative energies. On the other hand, DFT 18 methods often fail to reproduce excited-state properties and non-bonded interaction such as hydrogen bonding and van der Wals interactions. Time-dependent density functional theory (TDFT) 30,31 has been introduced to calculate excitation energies and oscilator strengths. In this case, the Kohn-Sham (KS) orbital energies and various exchange integrals are used in place of matrix elements of the Hamiltonian. TDFT is usualy most succesful for low-energy excitations, because the KS orbital energies for orbitals that are high in the virtual manifold are typicaly quite poor. Further developments aimed at correcting systematic erors in TDFT offer great promise for future applications. 1.7 Kinetics The study of chemical kinetics is a fundamental part of chemistry. Chemistry is the study of reactions; therefore determination of the rates is important. The results of kinetic studies give information which can be applied in diferent ways. For example many important phenomena, such as combustion or stratospheric ozone depletion, involve many reaction steps. If we want to understand these proceses fully, determination of the rate of each step is necesary. The advantages of such knowledge would be significant such as in developing more eficient combustion proceses or reducing ozone depletion. Recent developments in experimental techniques and computational methods have alowed detailed studies of elementary reaction rates. The variation of such rates with temperature or presure can give microscopic insights into the molecular mechanisms of these reactions. 19 1.7.1 Bimolecular Reactions One of the basic aims of theoretical reaction kinetics is to understand why some reactions are fast and others slow, why some reactions have strong positive temperature dependencies and others none. Obviously we need to compare our theoretical expresions with experimental data or empirical expresions for the temperature dependence of the rate coeficient. Initialy we wil limit ourselves to trying to match the Arhenius expresion 32 [k=Aexp(-E a /RT]. Rate constant, Arhenius constant, activation energy, and temperature are denoted by k, A, E a , T, respectively. Although this expresion is not perfect, it is a good starting point for kinetics. First of al I wil start with the collision theory 3 in which the molecules are thought of as hard, structureles spheres. Then a more detailed theory which is statistical, rather than collisional, wil be described. This is the famous transition state theory 3 (TST) which was developed in the 1930s and has ben used in much of the discussion of rate proceses. 1.7.1.a Collision theory Collision theory is a pictorialy simple model that provides a good initial visualization of bimolecular reaction. It emphasizes the importance of collision events providing the energy for reaction, and predicts qualitatively the form of temperature dependence of the rate coeficient. The predicted values for A are far from the experimental results. There are some disadvantages in collision theory. Firstly, the 'hard sphere' asumption has completely ignored the structure of the molecules. Secondly, it is asumed that molecules react instantaneously. In practice the 20 changes in structure take place over a finite period. The structure of the reaction complex wil evolve and this must be considered. Finaly it is asumed that there are no interactions betwen molecules until they come into contact. Because of these drawbacks, we must turn to transition state theory (TST) to make further progres. 1.7.1.b Transition State Theory Once simple collision theory was found to be inadequate, another was developed in the 1930s, initialy by Wigner and Pilzer, later extended by Eyring and known as activated complex theory or now more commonly as transition state theory. Transition state theory refers to the details of how reactions become products. In order for a reaction to occur, the transition state must pas through some critical configuration in this space. Because of the nature of the potential function used to expres the energy of the system as a function of atomic positions, the system energy posseses a saddle point. The esential feature of the transition state theory is that there is a "concentration" of the species at the saddle point, the activated complex, that is in equilibrium with reactants and products. The Boltzmann distribution governs the concentration of that activated complex, and the rate of reaction is proportional to that concentration. Since the concentration of activated complex is smal because its energy is higher than that of the reactants, this critical configuration represents the regulator of the rate of flow of reactants to products. The concentration of the activated complex is not the only factor involved because the frequency of its disociation into products plays an important role. Therefore, the rate can be expresed in eq 17. 21 ! Rate= Activated complex concentration " # $ % & ? Decomposition frequency of the activated complex " # $ % & ? (17) In order for the activated complex to separate into products, one bond (the one being broken) must acquire sufficient vibrational energy to separate. When it separates, one of the 3N-6 vibrational degres of fredom is lost and is transformed into translational degres of fredom of the products. Central to the ideal of transition state theory is the asumption that the activated complex is in equilibrium with the reactants (eq 18). A +B [AB] Products (18) ? For the formation of the activated complex, [AB] ? , the equilibrium constant is given in eq 19. K = [AB] (19) [][] ? ? Reaction rate can be writen in terms of the equilibrium constant as eq 20. Rate= kT h K[A][B] (20) ? 22 Equilibrium constant K ? is also equal to exp(-?G ? /RT). Therefore the rate constant is given in eq 21. ! k= kT h K ? = kT h exp("#G ? /RT) (21) 1.7.1.c Variational Transition State Theory Transition state theory is exact if and only if no trajectories cross the transition- state dividing surface more than once. When the transition state theory is not exact, the transition state theory overestimates the exact (equilibrium) rate constant. One should therefore pick the transition-state dividing surface to minimize the flux through it. Variational transition state theory 34-37 (VTST) has been developed in which the position of the transition state is varied until the maximum value of ?G ? is found. Note that VTST always provides an equal or a beter estimate of the rate constant than TST. 1.8 Unimolecular Reactions A unimolecular reaction 38 is in principle the simplest kind of elementary reaction, since it involves the isomerization or decomposition of a single isolated reactant molecule (A) through an activated complex (A ? ) which involves no other molecule (eq 22). A ? A ? ? Products (22) 23 At the beginning of the twentieth century, many gas-phase reactions were known to be first-order proceses and were thought to be first order under al conditions. Many reactions such as pyrolyses of simple ketones, aldehydes and ethers, have been found not to be unimolecular proceses acording to the modern definition. Despite the complexity, the earlier studies of these reactions were important in the development of unimolecular reaction theory. It was dificult to understand how first order proceses could result if molecules were energized by bimolecular collisions that would be expected to be second order proceses. Lindemann explained this phenomena by predicting that the rate coeficient should decrease with presure, with the reaction eventualy becoming second order overal. Lindemann theory 39 overcomes the disadvantages of earlier theories. The four main concepts of the theory are given below (a-c) (a) By collisions, a certain fraction of the molecules become energized. The rate of the energization proces depends upon the rate of bimolecular collisions. This proces can be writen as eq 23 where A + M A * + M (23) k 1 M represent a product molecule, an added 'inert' gas molecule, or a second molecule of reactant. In the simple Lindemann theory k 1 is energy-independent. (b) Energized molecules are de-energized by collision. This is the reverse of proces eq 23 and may be writen as eq 24. 24 A * + M k 2 A + M (24) The rate constant k 2 is energy-independent. The superscript * indicates the energized species. (c) The unimolecular disociation proces eq 24 also occurs with a rate constant independent of the energy content of A * (eq 25). A * B+C (25) k 3 Before starting to RKM theory, we wil explain the definition of partition functions which are important in the RKM theory. If a molecular system can be writen in a series of quantized energy levels with total energy E 0 , E 1 , E 2 , .., then the partition function Q for the molecule is defined as in eq 26. Q = g 0 exp(-E 0 /kT) + g 1 exp(-E 1 /kT) + g 2 exp(-E 2 /kT) + .. (26) = ! g i i=0 " # exp($E i /kT) where g i is the degeneracy of the energy level E i . The degeneracy is alternatively the number of diferent independent wave-functions of the system with total energy E i or the number of physicaly distinct ways of the energy can be distributed in the molecule. Therefore the sum can be taken over al energy levels rather than evaluating ?exp(-E i /kT) over al quantum states (some of the same energy), which is mathematicaly more 25 complicated. For a given system, Q depends only on the temperature and on the zero chosen for the energy scale. If the energy zero is shifted down by an amount ?E al the energies are increased by ?E. Therefore each exponential term in eq 26 is multiplied by exp(-?E/kT) and Q is simply multiplied by the same factor. Partition functions can be considered for certain degres of fredom of the molecule rather than for a whole molecule. It is comon to consider separately the electronic, vibrational, rotational and translational partition functions, Q e , Q v , Q r and Q t . If the total energy can be writen as E tot = E e + E v + E r + E t , the molecular partition function is the product of the individual functions (Q= Q e ?Q v ?Q r ?Q t ). In the same way al the degres of fredom of one type do not need to be considered together. For instance, Q v can be factorized into contributions for diferent vibrations. Such an approach may be useful if an approximate treatment of Q v is valid for some vibrations but not for others. 1.8.1 RKM Theory This theory was developed by R.A. Marcus 40 and O.K.Rice 41 and is known by the names of these authors or very often by the initials RKM. In RKM theory, the main diference from the previous theories is the calculation of the rate constant k 1 by quantum statistical mechanics. The reaction scheme used in the RKM theory comprises the reactions shown in eq 27 and eq 28. A + M dk 1(E * ->E * +dE * ) k 2 A * (E->E * +dE * ) + M (27) A * (E) k ! (E * ) A k Products (28) ? ? 26 In this theory k 1 is evaluated as a function of energy by a quantum statistical- mechanical treatment instead of the clasical treatment. E * is the total non-fixed energy in the active degres of fredom of a given energized molecule A * (E * = E ? v + E * r ; E ? v and E * r are the vibrational and rotational part of E * , respectively). In this scheme, a careful distinction has been made betwen the energized molecule A * (sometimes caled active molecule) and the activated complex A ? (occasionaly caled the activated molecule). The energized molecule A * is basicaly an A molecule which is characterized loosely by having enough energy to react. The energy distribution wil not usualy be such that reaction occurs imediately. However, there wil be many quantum states of the energized molecule in a given smal energy range and only a few of these can undergo conversion to products. Moreover, the energized molecules wil not react instantaneously even when one of these relatively rare quantum states is reached, since the vibrational modes involved in the reaction wil not be in the correct phase at first. The energized molecules thus have lifetimes to decomposition which are much greater than the periods of their vibrations. The actual lifetimes to de-energization or decomposition depend on the values of k 2 [M] and k ? [E * ] respectively. They are typicaly in the range 10 -9 to 10 -4 s. The activated complex A ? is basicaly a species which is recognizable as being intermediate betwen reactant and products. It is characterized by having a configuration corresponding to the top of an energy barier betwen reactant and products. The energy profile along the reaction coordinate involves a potential energy barier E 0 (the critical energy requirement) betwen reactant and products and this must be surmounted for 27 reaction to occur. The activated complex is a molecule which lies in an arbitrarily smal range at the top of the barier. The activated complex is thus unstable to movement in either direction along the reaction coordinate and, in contrast to an energized molecule, has no measurable life. The unimolecular rate constant for RKM can be expresed as in eq 29. ! kuni= L " Q1 + hQ12 {P(E vr + )}exp(#E * /kT)dE * $ 1+k % (E * )/k 2 [M] E * =E 0 & ? (29) L ? is the statistical factor which concerns the possibility of a reaction that can proced by several distinct paths which are kineticaly equivalent. For example the disociation of H 2 O to OH+H, for which L ? =2 since either of the two identical OH bonds are broken. Q 1 ? and Q 1 are the partition functions for the adiabatic rotations in the activated complex and the A molecule respectively. Q 2 is the partition function for the active degres of fredom of the reactant molecule A. E ? is the total non-fixed energy in the active degres of fredom of a given activated complex A ? (E ? = E ? v + E r ? + x; x is the translational energy of A ? in the reaction coordinate E ? v and E ? r are the vibrational and rotational contributions to E + ). ?P(E ? vr ) is the sum of the numbers of vibrational-rotational quantum states at al the quantized energy levels of energy les than or equal to E ? . It is simply the total number of vibrational-rotational quantum states of the activated complex with energies ?E ? Planck and Boltzmann gas constants are denoted by h and k, respectively. 28 1.8.2 The High Presure Limit The high-presure limit is easily obtained from eq 29 by putting [M] ? ?, when the k uni becomes the presure-independent first-order rate constant k ? . The rate constant at high presure can be expresed in eq 30. ! k"=L # kT h Q ? exp($E 0 /kT) (30) Q and Q ? are the complete vibrational-rotational partition functions for the reactant and the activated complex (Q=Q 1 Q 2 and Q ? =Q 1 ? Q 2 ? ). 1.8.3 The Low-Presure Limit In the limit of very low presures the first-order rate constant from eq 29 becomes proportional to the presure. The rate constant at the low presure (k bim ), that is the second-order rate constant, is given by eq 31, in which Q 2 * is the partition function for energized molecules (specificaly those of A molecules which have non-fixed energy greater than E 0 ) using the ground state of A for the zero of energy. ! k bim = [M]">0 lim ( k uni [M] )= k 2 Q 2 * 2 (31) 29 Equation 31 can be writen in terms of the partition function Q 2 * ' for the ground state of energized molecules. The two partition functions are related by the equation Q 2 *' = Q 2 * exp(E 0 /kT) and k bim is given by eq 32. The terms k 2 exp(E 0 /kT) in this ! k bim = Q 2 *' 2 k 2 exp("E 0 /kT) (32) equation correspond to Lindemann's k 1 . The density of quantum states increases rapidly with energy and Q 2 *' is therefore greater than Q 2 . It has already been sen that the Arhenius activation energy of a unimolecular reaction varies with presure. The RKM theory does not lead to any simple equation for this variation, but the theoretical activation energy E ARR at any presure may be obtained in the usual way from the first-order rate constants calculated at a series of temperatures. The basic reason for the variation of E ARR with presure is the change in the energy distribution of reacting molecules. At sufficiently low presures, al the molecules which become energized react. The rate constant k 1 decreases rapidly as the energy increases. Therefore the reaction of molecules with energies near the critical energy is favored. At high presures there is a competition betwen reaction and collisional de-energization. The energized molecules with energies near the critical energy have long lifetimes before reaction. Therefore they can be de-energized rather than rapidly reacting with molecules having higher energies. 30 1.8.4 Adiabatic Rotations An adiabatic rotation means that the angular momentum stays constant during the conversion of the energized molecule to an activated complex. In another way, the rotation stays in the same quantum state throughout this proces. Since the energy of the rotation is given by E J =(h 2 /8? 2 I)J(J+1), the energy wil change as the geometry of the molecule and hence the moment of inertia I changes. In most cases where such efects are worth considering, I ? >I so that E J >E J ? and the adiabatic rotations release energy into the other (active) degres of fredom of the molecule. This increases the multiplicity of available quantum states of the complex and the specific rate constant k ? . An alternative interpretation is that this 'centrifugal efect' alows part of the adiabatic rotational energy to be used for overcoming the potential energy barier, thus efectively reducing E 0 . In bond fision reactions the moment of inertia can change substantialy, amounting to an efective reduction of E 0 by more than kT and increase in k ? by more than a factor of e. 31 1.9 References (1) Jensen F. Introduction to computational Chemistry, Wiley: New York, 1999. (2) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio Molecular Orbital Theory, Wiley and Sons: New York, 1986. (3) Levine, I. N. Quantum Chemistry, Fifth Ed., Prentice-Hal, Inc.; New Jersey, 2000. (4) Thiel, W. Tetrahedron 1988, 44, 7393. (5) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. (6) (a) Hehre, W. J.; Stewart, F. R.; Pople, J. A. J. Chem. Phys. 1969, 51, 2657. (b) Newton, M. D.; Lathan, W. A.; Hehre, W. J.; Pople, J. A. J. Chem. Phys. 1969, 51, 3927. (c) Hehre, W. J.; Pople, J. A. J. Am. Chem. Soc. 1970, 92, 2191. 7) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864. 8) Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc. 1977, 99, 4499. 9) Bingham, R. C.; Dewar, M. J. S. J. Am. Chem. Soc. 1975, 97, 1285. 10) Stewart, J. J. P. J. Comp. Chem. 1989, 10, 209. 11) Stewart, J. J. P. J. Comp. Chem. 1989, 10, 221. 12) Bartlet, R. J. Ann. Rev. Phys. Chem. 1981, 32, 359. 13) Krishnan, R.; Schlegel, H. B.; Pople, J. A. J. Chem. Phys. 1980, 72, 4654. 14) Raghavachari, K.; Pople, J. A. Int. J. Quant. Chem. 1981, 20, 167. 15) Foresman, J. B.: Frisch, A. Exploring Chemistry with Electronic Structure Methods, 2nd Ed., Gaussian Inc.: Pitsburgh, 1996. 32 16) Slater, J. C. Phys. Rev. 1930, 36, 57. 17) Cramer, C. J. Esentials of Computational Chemistry, John Wiley & Sons Inc.: New York, 2002. 18) Hay, P. J.: Wadt,W. R. J. Chem. Phys. 1985, 82, 270. 19) Stevens, W. J.; Basch, H.; Krauss, M. J. Chem. Phys. 1984, 81, 6026 20) Dolg, M.; Wedig, U.; Stoll, H.; Preuss, H. J. Chem. Phys. 1987, 86, 866. 21) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Int. J. Quant. Chem. 1978, 14, 545. 22) Bernardi, F.; Bottini, A.; McDougal, J. J. W.; Robb, M. A.; Schlegel, H. B. Far. Synp. Chem. Soc. 1984, 19, 137. 23) Cizek, J. Adv. Chem. Phys. 1969, 14, 35. 24) Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J. Chem. Phys. 1987, 87, 5968. 25) Le, T. J.; Taylor, P. R. Int. J. Quantum Chem. Symp. 1989, 23, 199. 26) Par, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules, Oxford Univ. Pres: Oxford, 1989. 27) Le, C.; Yang, W.; Par, R. G. Physical Review B, 1988, 37, 785. 28) Miehlich, B.; Savin, A.; Stoll, H.; Preus, H. Chem. Phys. Let. 1989, 157, 200. 29) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. 30) Stratmann, R. E.; Scuseria, G. E.; Frisch, M. J. J. Chem. Phys. 1998, 109, 8218. 31) Bauernschmit, R.; Ahlrichs, R. Chem. Phys. Let. 1996, 256, 454. 32) Arhenius, S. Zeitschrift fur Physikalische Chemie 1889, 4, 226. 33) Weston, R. A, Jr.; Schwarz, H. A. Chemical Kinetics, Prentice-Hal, Inc., Englewood Clifs: New Jersey, 1972. 33 34) Truhlar, D. G.; Garet, B. C. Ac. Chem. Res. 1980, 13, 440. 35) Bishop, D. M.; Laidler, K. J. Trans. Faraday Soc. 1970, 66, 1685 36) Keck, J. C. J. Chem. Phys. 1960, 32, 1035. 37) Keck, J. C. Adv. Chem. Phys. 1967, 13, 85. 38) Forst, W. Unimolecular Reactions, Cambridge University Pres: Cambridge, 2003. 39) Lindemann, F. A. Trans. Faraday Soc., 1920, 17, 598 40) Marcus, R. A. J. Chem. Phys. 1952, 20, 359. 41) Marcus, R. A.; Rice, O. K. J. Phys. and Colloid Chem. 1951, 55, 894. 34 CHAPTER 2 COMPUTATIONAL STUDY OF THE REACTIONS BETWEEN XO (X=Cl, Br, I) AND IMETHYL SULFIDE 2.1 Introduction Dimethyl sulfide (DMS) is the most important reduced sulfur compound in the troposphere where it plays a major role regulating the climate through formation of cloud condensation nuclei in the troposphere. 1 Therefore, it is of great importance to understand the mechanism of DMS atmospheric transformations. 2-4 Unfortunately, there are many uncertainties in understanding the eventual fate of DMS in the atmosphere because of the lack of understanding of the oxidation mechanism. DMS, which is naturaly emited by oceanic phytoplankton, can reach the troposphere and react with halogen monoxide radicals in the catalytic ozone-loss cycle (eq 1). 5 DMS + XO ? DMSO + X (X=Cl, Br, I) (1a) O 3 + X ? O 2 + XO (1b) A beter understanding of the reactions of XO with DMS wil permit a quantitative asesment of the importance of these reactions in atmospheric models. 35 The experimental measurements of the reaction rates betwen XO and DMS (X=Cl, Br, I) are collected in Table 1. 6-16 While the hydrogen abstraction pathway is the major pathway in the reaction of Cl and NO 3 with DMS, this pathway does not appear to be important in the reaction of XO (X=Cl, Br, I) with DMS. Instead, only the oxygen- atom transfer (OAT) pathway is observed, with DMSO as the single product detected in high yield. As sen from Table 1, there is fairly good agrement on rate constants of ClO and BrO reactions with DMS. On the other hand, the rate constant varies by almost four orders of magnitude for the IO + DMS reaction. At the slow end, the reaction would not be an important atmospheric loss proces for DMS, while it would be on the fast end. One of the goals of the present study is to determine the rate constant for the IO + DMS reaction as acurately as possible. From the experimental exothermicities 17 for the reactions of ClO, BrO and IO with DMS to form Cl, Br, and I plus DMSO (-22.4 kcal?mol ?1 , ?27.5 kcal?mol -1 , and ? 29.0 kcal?mol -1 , respectively), one would expect the reactivity order ClO < BrO < IO. The reactions of ClO and BrO follow this trend, but the variations in the rate constant determinations for the IO reaction are too large to determine whether the trend is followed. We wil consider two possible reaction pathways; (a) oxygen-atom transfer (OAT) from XO (X=Cl, Br, I) to DMS and (b) abstraction of hydrogen by XO (X=Cl, Br, I) (eq 2-3). MeSCH 3 + XO ? MeS(O)CH 3 + X (2) 36 MeSCH 3 + XO ? MeSCH 2 + HOX (X=Cl, Br, I) (3) For X=Cl, an additional pathway is considered, the abstraction of hydrogen from DMSO by chlorine atom (eq 4) which is exothermic for X=Cl but Table 1. Experimental and theoretical rate constant (cm 3 ?s -1 ) for XO + DMS (X=Cl, Br, I) at 298K Presure(torr) Exptl. k Ref Calc. k a ClO+DMS Flow tube MS 1.8-5.1(He) (9.5?2.0)x10 -15 6 3.0x10 -15 Flow tube S 0.5-2 (3.9?1.2)x10 -15 7 BrO+DMS b Flow tube MS 1.8-5.1(He) (2.7?0.5)x10 -13 6 8.7x10 -13 Flow tube S (2.7?0.2)x10 -13 8 Laser photolysis absorption 60-100(N 2 ) (4.4?0.6)x10 -13 9 Cavity ring down spectroscopy 100(N 2 ) 4.2x10 -13 10 IO+DMS Smog chamber FTIR 760(N 2 ) (3?1.5)x10 -1 11 1.5x10 -1 Flow tube MS 1(Ne) (1.5?0.5)x10 -1 12 Laser photolysis absorption 40-300 (N 2 ,O 2 ,air) <3.5x10 -14 13 Flow tube MS 1.58-5.1(He) (8.8?2.1)x10 -15 6 Flow tube S 1.1-1.4(He) (1.5?0.2)x10 -14 14 Flow tube MS 2.5-2.7(He) (1.6?0.1)x10 -14 15 Cavity ring down spectroscopy 100(N 2 ) (2.5?0.2)x10 -13 16 a This work. b The rate constant for the BrO + MeSH reaction at 298K is 4.54x10 -14 cm 3 ?s -1 . Aranda, A.; D?az de Mera, Y.; Rodr?guez, D.; Salgado, S.; Mart?nez, E. Chem. Phys. Let. 2002, 357, 471. 37 MeS(O)CH 3 + Cl ? MeS(O)CH 2 + HCl (4) endothermic for X=Br and I. 2.2 Method Al calculations were made using the Gaussian 98 program 18 system. Optimization and frequency calculations were caried out at the B3LYP/6-311+G(d,p) level (X=Cl and Br) and B3LYP/6-311+G(d,p)/ECP (X=I). Al imaginary frequencies for transition states were tested by using the graphical program (MOLDEN) 19 to make sure that the motion was appropriate for converting reactants to products. For X=Cl, G3B3 20 energies have been determined using B3LYP/6-31G(d) geometries. For X=Br and I, a series of single-point calculations (using B3LYP/6-311+G(d,p) or B3LYP/6- 311+G(d,p)/ECP geometries) have been combined to form G2B3(MP2) 21,2 and G3B3(MP2) 2,24 energies with only slight deviations from the standard procedure. The G3B3(MP2) energies were not determined for X=I because the method has not been defined for the iodine atom. G2B3(MP2) theory corresponds efectively to calculations at the QCISD(T)/6- 311+G(3df,2p) level with zero-point vibrational energies (ZPE) and higher-level corrections (HLC). The G2B3(MP2) method was extended by Radom and co-workers 2 to bromine- and iodine-containing compounds using basis sets which include efective core potentials (ECP) 25 and first-order spin-orbit corrections (SOC) for Br and I atoms (?ESO). Recently, the G3B3(MP2) method has been extended to X=Br where al- electron basis sets are used on bromine. 24 In the calculations presented here (and a slight deviation from the standard G2B3(MP2) and G3B3(MP2) methods), SOC wil be 38 included for X, XO, XO/DMS complexes and XO/AT transition states (X=Cl, Br, I). The XO radicals have 2 ? ground states and are expected to have large SOCs. The XO/DMS complexes (and to a leser extent XO/AT transition states) have weak interactions betwen the XO radical and DMS such that the spin-orbital coupling efect may not be fully quenched. The SOCs have been calculated for X, XO, XO/DMS complexes and XO/AT transition states (X=Cl, Br, and I) and compared with other calculations and experiment in Table 2. The calculations used full Breit-Pauli spin-orbit coupling, 26 an al-electron 6- 311G(d) basis set, and an active space that varied with the compound (5e,3o), (13e,8o), and (15e,10o) for X, XO and XO-DMS/XO-OAT-TS, respectively). The calculated and experimental SOCs are in reasonable agrement for X and XO (Table 2) which suggests that the XO-DMS/XO-OAT-TS SOCs may have similar acuracy. We have decided to use the experimental SOCs for X and XO and the calculated SOCs for XO-DMS/XO- OAT-TS. While the SOC for BrO and IO are large (0.92 and 1.99 kcal?mol -1 , respectively) and should not be ignored, the SOCs for XO-DMS and XO-OAT-TS species are al les than 0.13 kcal?mol -1 and could probably be ignored. Morokuma and co-workers 27 have computed SOCs at a similar level of theory for reactions including IO + C 2 H 5 ? HOI + C 2 H 4 at the G2M(RC) level and have obtained similar corrections. Relative energies (and related thermodynamic properties) are given in Table 3. The structures and enthalpies of species on the XO-DMS PES are given for ClO-DMS in Figures 1 and 2, for BrO-DMS in Figures 3 and 4 and for IO-DMS in Figures 5 and 6. Rate constants were calculated by using CHEMRATE software package. 28 Vibrational frequencies and structures were calculated at the B3LYP/6-311+G(d,p) level 39 for X=Cl and Br and at B3LYP/6-311+G(d,p)/ECP for X=I and used as input along with energies at the G3B3, G3B3(MP2), and G2B3(MP2) levels for X=Cl, Br, and I, respectively. The normal modes corresponding to methyl and XO rotation were treated Table 2. Spin-orbit Corrections a (SOC) in kcal?mol -1 Calculated Exptl. this work b others Cl 0.78 0.82 c 0.84 d ClO 0.33 0.20 e ,0.30 f ,0.28 g 0.30 h ClO-SMe 2 -c C s 0.01 ClO-OAT-TS-c C s 0.00 Br 3.20 3.63 i 3.51 d BrO 0.84 0.64 e ,0.92 i ,0.78 g 0.92 j BrO-SMe 2 -c C s 0.05 BrO-OAT-TS-c C s 0.02 I 6.55 7.02 k 7.25 l IO 1.48 1.60 k ,1.70 e ,1.75 g 1.99 m IO-SMe 2 -c C s 0.13 IO-OAT-TS-c C s 0.08 a The Spin-Orbit Correction (SOC) is the diference betwen the lowest spin-orbit coupled state and the J-averaged state. For the 2 P and 2 ? electronic states of X and XO (X=Cl, Br, I), respectively, the SOC is ?/2 where ? is the Spin-Orbit Coupling Constant (SOC). Equivalently, the SOC is 1/3 of the Fine Structure Spliting (FS) where FS=?( 2 P 1/2 - 2 P 3/2 ) for X and FS=?( 2 ? 1/2 - 2 ? 3/2 ) for XO, respectively. b The GAMES program was used to calculated the full Breit-Pauli spin-orbit coupling 26 with a (5e,3o), (13e,8o), and (15e,10o) active space for X, XO, and XO-SMe 2 , respectively and a 6-311G(d) al-electron basis set. GAMES: Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. J.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery J. A. J. Comput. Chem. 1993, 14, 1347. Al-Electron 6- 311G(d) basis set: Krishnan, R.; Binkley, J. S.; Seger R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650. McLean A. D.; Chandler, G. S. J. Chem. Phys. 1980, 72, 40 5639. Blaudeau, J.-P.; McGrath, M. P.; Curtis, L. A.; Radom, L. J. Chem. Phys. 1997, 107, 5016. Curtis, L. A.; McGrath, M. P.; Blandeau, J. P.; Davis, N. E.; Binning, R. C., Jr.; Radom, L. J. Chem. Phys. 1995, 103, 6104. Glukhovstev, M. N.; Pross, A.; McGrath, M. P.; Radom, L. J. Chem. Phys. 1995, 103, 1878. c Ilias?, M.; Kel?, V.; Vischer, L.; Schimelpfenning, B. J. Chem. Phys. 2001, 115, 9667. d Moore, C. E. Atomic Energy Levels (National Bureau of Standards, Washington, D.C. 1971), Vols. I and I, NSRDS-NBS 35. e Ma, N. L.; Cheung, Y.-S.; Ng, C. Y.; Li, W.-K. Mol. Phys. 1997, 91, 495. f Berning, A.; Schwqeizer, M.; Werner, H.-J.; Knowles, P. J.; Palmieri, P. Mol. Phys. 2000, 98, 1823. g Koseki, S.; Gordon, M. S.; Schmidt, M. W.; Matsunaga, N. J. Phys. Chem. 1995, 99, 12764. h Coxon, J. A. Can. J. Phys. 1979, 57, 1538. i Bladeau, J.-P.; Curtis, L. A. Int. J. Quantum Chem. 1997, 61, 943. j McKelar, A. R. W. J. Mol. Spectrosc. 1981, 86, 43. k Roszak, S.; Krauss, M.; Alekseyev, A. B.; Liebermann, H.-P.; Buenker, R. J. J. Phys. Chem. A 2000, 104, 2999. l Lias, S. G.; Bartmas, J. E.; Liebman, J. F.; Holmes, J. L.; Levin, R. D.; Malard, W. G. J. Phys. Chem. Ref. Data 17, Suppl. No 1 1988. m Giles, M. K.; Polak, M. L.; Lineberger, W. C. J. Chem. Phys. 1991, 95, 4723. as torsions rather as vibrations and the experimental spin-orbit spliting of XO was included as input to the CHEMRATE program. Natural bond orbital (NBO) analysis 29 was done at the B3LYP/6-311+G(d,p) level for X=Cl and Br and at the B3LYP/6-311+G(d,p)/ECP level for X=I to understand the nature of bonding in the XO-DMS complexes. 41 Table 3. Relative Energies (kcal?mol -1 ) at the DFT and ab initio levels for various species involved in the XO + DMS Reactions (X=Cl, Br, I) B3LYP/6-31+G(d,p) ab initio a E e ?H(0K) ?H(298K) ?G(298K) ?H(0K) ?H(0K) +SOC ?H(298K) ?G(298K) ClO+DMS 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ClO-SMe 2 -t C 1 -7.2 -6.3 -6.1 2.1 -1.6 -1.4 -1.2 7.1 ClO-SMe 2 -t C s -7.1 -6.3 -6.7 3.2 -1.5 -1.3 -1.6 8.3 ClO-SMe 2 -c C s -6.1 -5.3 -5.2 3.3 -1.8 -1.6 -2.0 6.5 ClO-abst-complex -5.2 -5.4 -5.0 1.6 -7.0 -6.8 -6.5 0.1 ClO-OAT-TS-t C s 1.4 1.7 1.5 10.8 7.2 7.4 7.3 16.6 ClO-OAT-TS-c C s -1.5 -1.0 -1.1 8.9 1.5 1.7 1.3 1.3 ClO-abst-TS 2.4 -0.2 -0.6 8.5 2.3 2.5 2.1 1.2 MeS(O)Me+Cl -12.8 -1.8 -12.0 -8.8 -2.2 -2.8 -23.0 -19.8 MeSCH 2 +HOCl 0.5 -1.6 -0.9 -3.7 -2.2 -12.0 -1.6 -4.3 MeS(O)CH 2 +HCl -9.3 -12.9 -12.4 -1.7 -24.7 -24.5 -23.9 -23.2 Cl-abst-TS b -12.1 -14.3 -15.0 -4.7 -23.3 -2.9 -23.8 -13.4 BrO+DMS 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 BrO-SMe 2 -t C 1 -6.8 -6.3 -6.1 2.1 -2.8 -1.9 -1.7 6.5 BrO-SMe 2 -t C s -6.8 -6.4 -6.1 1.9 -2.5 -1.6 -1.4 6.6 BrO-SMe 2 -c C s -5.4 -5.0 -4.8 3.4 -2.8 -1.9 -1.7 6.5 BrO-abst-complex -6.4 -6.9 -6.5 0.8 -10.0 -9.1 -8.7 -1.4 BrO-OAT-TS-t C 1 -0.7 -0.6 -0.8 8.7 3.2 4.1 3.9 13.4 BrO-OAT-TS-t C s -0.7 -0.7 -1.4 9.5 3.1 4.0 3.4 14.2 BrO-OAT-TS-c C s -2.7 -2.5 -2.8 7.1 -1.1 -0.2 -0.4 9.4 BrO-abst-TS 1.9 -1.0 -1.3 8.1 1.3 2.3 2.0 1.4 MeS(O)Me+Br -15.9 -15.0 -15.2 -1.9 -27.7 -30.3 -30.5 -27.2 MeSCH 2 +HOBr -1.2 -3.7 -3.0 -5.7 -5.6 -4.7 -4.0 -6.7 IO+DMS 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 IO-SMe 2 -t C 1 -8.9 -8.0 -7.8 0.4 -3.4 -1.5 -1.3 6.9 IO-SMe 2 -t C s -8.7 -7.9 -7.7 -0.3 -2.6 -0.8 -0.5 6.8 IO-SMe 2 -c C s -6.6 -6.0 -5.7 2.5 -2.2 -0.4 -0.1 8.1 IO-abst-complex -10.9 -1.2 -10.8 -3.8 -13.3 -1.4 -10.9 -4.0 IO-OAT-TS-t C 1 -6.8 -6.3 -6.4 2.6 -1.1 0.8 0.7 9.8 IO-OAT-TS-t C s -6.7 -6.4 -7.0 3.8 -1.6 0.4 -0.3 10.6 IO-OAT-TS-c C s -6.6 -6.0 -6.2 3.3 -4.4 -2.4 -2.6 6.9 IO-abst-TS -0.3 -2.8 -3.1 5.9 1.6 3.6 3.3 12.3 MeS(O)Me+I -25.8 -24.4 -24.6 -21.3 -32.9 -38.1 -38.3 -35.0 MeSCH 2 +HOI -5.3 -7.7 -7.0 -9.8 -8.5 -6.5 -5.8 -8.6 42 Footnotes for Table 3. a The values for ClO+DMS, BrO+DMS, IO+DMS at the ab initio level are calculated by using G3B3, G3B3(MP2) and G2B3(MP2) methods, respectively. The G3 methods include SOC for atoms heavier than Ne. The entries under ?H(0K) have this corection subtracted. The entries under ?H(0K)+SOC have SOC added for X and XO as wel as the XO/DMS complexes and OAT transition states. The SOC of X and XO are from experiment and the SOC of XO-SMe 2 -c (C s ) and XO-OAT-TS-c (C s ) are from calculations. The SOC of XO-SMe 2 -t and XO-OAT-TS-t (trans orientation) was asumed to be the same as the corresponding cis structure. The thermodynamic properties are obtained from unscaled vibrational frequencies. b The transition state ?Cl-abst-TS? was located at the MP2/6-31G(d) level because the structure could not be located at the DFT level. Further calculations (DFT and ab initio) were made on this geometry. 2.3 Results and discusion 2.3a ClO+DMS: Two orientations of ClO with DMS were considered, cis and trans. At the B3LYP/6-311+G(d,p) level, the trans ClO-DMS complex (C s complex) was 1.0 kcal?mol -1 more stable than the cis complex. The C s trans complex had one imaginary frequency at the B3LYP/6-311+G(d,p) level and led to a C 1 minimum, 0.1 kcal?mol -1 lower in energy. At the G3B3 level, the cis complex was 0.2 kcal?mol -1 lower in energy than the trans complex (0.8 kcal?mol -1 lower with SOC at 298K). In addition, the binding enthalpy of the complex is much smaler at the G3B3 level (2.0 kcal?mol -1 ) compared to DFT (6.7 kcal?mol -1 ). The oxygen-atom transfer (OAT) transition states were also computed in the cis and trans orientations where the former was lower in enthalpy than the later by 6.0 43 kcal?mol -1 (G3B3, Table 3). While the structures of the two transition states (ClO-OAT- TS-c and ClO-OAT-TS-t) are similar, the cis TS is earlier than the trans as judged by the longer S-O distance (1.985 compared to 1.964 ?) and the shorter Cl-O distance (1.861 compared to 1.901 ?), which is consistent with the smaler activation enthalpy (3.3 compared to 8.9 kcal?mol -1 ). The lower enthalpy of the cis TS (relative to the trans TS) is due to the electrostatic stabilization of transfering charge on the ClO unit with the methyl hydrogens of DMS. The cis TS is also favored in XO-OAT-TS (X=Br and I) for the same reason, but the diference decreases from 6.0 kcal?mol -1 for X=Cl to 3.8 and 2.3 kcal?mol -1 for X=Br and I, respectively. The initial product of the OAT reaction is Cl + DMSO which is calculated to be 23.0 kcal?mol -1 more stable than ClO + DMS (22.4 kcal?mol -1 , exptl). However, the calculated bond disociation enthalpy of Cl-H and MeS(O)CH 2 -H are 103.0 kcal?mol -1 and 98.5 kcal?mol -1 , respectively, which indicates that the Cl radical can exothermicaly abstract a hydrogen from DMSO to form HCl + MeS(O)CH 2 . The transition state geometry for the abstraction was taken from MP2/6-31G(d) because the transition state could not be located at the B3LYP/6-31G(d) or B3LYP/6-311+G(d,p) levels. The enthalpy of Cl-abst-TS at G3B3 (using the MP2/6-31G(d) geometry) was 0.8 kcal?mol -1 below Cl + DMSO. On the other hand, the fre energy of Cl-abst-TS is 6.4 kcal?mol -1 higher than Cl + DMSO at 298K. Very similar results have been obtained by Vandresen et al. 30 where values of ?H ? =-2.5 kcal?mol -1 and ?G ? =4.6 kcal?mol -1 were obtained. In the abstraction transition state (ClO-abst-TS), the forming O-H bond is 1.265 ? and the breaking C-H bond is 1.290 ?. A hydrogen-bonded complex is formed (ClO- abst-complex) which is bound by 4.9 kcal?mol -1 relative to HOCl + MeSCH 2 . The smal 44 diference betwen the ClO-OAT-TS and the abstraction TS (1.3 compared to 2.1 kcal?mol -1 ) suggests that products from both pathways should be observed. However, the HOCl/MeSCH 2 products have not been reported from experiments. 6,7 In contrast to experimental results on the reaction betwen Cl + DMS, Stickel et al. 31 and Butkovskaya et al. 32 report that hydrogen abstraction is the dominant reaction pathway at low presure in the reaction betwen Cl + DMS. We note that the rate of H-abstraction from DMS by Cl is much faster than the rate of OAT from ClO to DMS (3.3?0.5)x10 -10 compared to (3.9?1.2)x10 -15 cm 3 ?s -1 at 298K, (Table 1). There are two ways to compute the rate constant; (1) as a bimolecular reaction with ClO and DMS as reactants and (2) as a unimolecular pathway with the ClO-DMS complex as reactant. In the first scenario, the A-factor wil be les favorable but the activation energy wil be smaler, while in the second scenario, the A-factor wil be more favorable but the activation energy wil be larger. In the unimolecular case, the rate constant was computed as k=K 5 ?k 6 where k 6 is obtained from CHEMRATE (eq 6) and K 5 is obtained from ?G=-RTlnK 5 using the calculated fre energy change of eq 5. ClO + DMS ? ClO-DMS (5) ClO-DMS ? Cl + DMSO (6) In the ClO + DMS reaction, the computed bimolecular rate constant was much faster than K 5 times the unimolecular rate constant (k 6 ). Thus, the enthalpy of binding of the ClO- DMS complex does not compensate for unfavorable entropy of binding and leads to K 5 much smaler than 1. Only the bimolecular rate constant is reported. 45 Cl O C C S 1.824 9. SMe 2 C C S 10.6 1.72 MeSCH 2 C O S Cl C 2.481 1.672 10.1 C 10.5 C O Cl S C 1.692.492 1.0C 10.4 C Cl O S C C 10.5 2.478 14.6 1.683 C Cl O S C 1.901 34.8 CS 10. 1.964 ClO-AT-S-t C s ClO-SMe 2 -t 1 ClO-SMe 2 -t C s ClO-SMe 2 -c C s C S C Cl O 1.2651.290 1.695 106.9 C 10.2 C S C O Cl 2.05 1.730 0.987 104. 10.8 ClO-abst-TS C 1 Cl-abst-complex C 1 ClO-AT-S-c C s Cl O 1.731 103.1 0.968 1.6 ClO HOCl 81. 94.6 95.8 C O S C Cl 1.287 1.508 1.845 C 9.0 108.2 MeS(O)CH 2 HCl C Cl O S C 1.861 28.4 97.1 C 10.2 1.85 CCl S O C 1.731.362 SC 10.5 Cl-abst-TS C 1 C O S C 1.5 C 96. 106.8 1.835 MeS(O)Me Figure 1. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p) level. Bond lengths are in ? and angles are in degres. The optimized structure of Cl-abst-TS is at the MP2/6-31G(d) level. 46 0. -2.0 1.3 2.1 -6.5 -1.6 -23.0 Cl O C S C C S O Cl C C Cl O S C C S C Cl O C S C + C O S C C S O C Cl OCl + + Cl DMS Cl-abst-TS ClO-AT--c HOCl MeS(O)Me Cl Cl-abst-complex ClO-AT-S-c C O S C + -23.9 MeS(O)CH 2 HCl MeSCH 2 Figure 2. Schematic diagram of the potential energy surface computed at the G3B3 (298K) level for the reaction of ClO with DMS. Relative energies are given in kcal/mol at 298 K 47 The calculated rate constant for the OAT pathway and abstraction pathway are 3.0x10 -15 and 1.1x10 -15 cm 3 ?s -1 , respectively at 298K and 760 tor; the branching ratio is 2.7 favoring the OAT pathway. The computed rate constant for the ClO + DMS reaction is in good agrement with experiment (Table 1). 2.3b BrO + DMS: The initial complex betwen BrO and DMS is 1.7 kcal?mol -1 more stable than separated reactants and nearly equaly stable in the cis and trans orientation. The nonbonded Br-H distance is 3.140 ? in the cis isomer (BrO-SMe 2 -c) is close to the sum of the van der Wals (vdW) radii of Br (1.97 ?) 3a nd H (1.10 ?) 3b which may indicate some interaction. In the corresponding ClO complex (ClO-SMe 2 -c), the nonbonding Cl-H distance is 2.894 ?, somewhat smaler than the sum of the vdW radii betwen chloride (1.90 ?) 3a nd hydrogen (1.10 ?). The transition state for OAT is 3.8 kcal?mol -1 lower in the cis than in the trans orientation due to more favorable electrostatic interactions in the former. Consistent with the smaler activation enthalpy, the cis OAT TS is earlier than the trans TS as judged by the longer forming S-O bond (2.013 compared to 1.972 ?) and the shorter breaking Br-O bond (1.964 compared to 2.004 ?). The activation enthalpy for the OAT pathway is negative with respect to separated reactants (-0.4 kcal?mol -1 ) while the fre energy of activation is 9.4 kcal?mol -1 . For comparison, Bedjanian et al. 8 and Nakano et al. 10 have determined negative activation energies of ?2.1?0.5 kcal?mol -1 and ?1.7?0.5 kcal?mol -1 , respectively for this reaction. The products, Br + DMSO, are calculated to be 30.5 kcal?mol -1 exothermic which can be compared to the experimental value 17 of 27.5 kcal?mol -1 . The Br radical is not expected to abstract a hydrogen from DMSO because the calculated Br-H bond enthalpy 48 BrO-SMe 2 -t C s BrO-AT-S-t C s BrO-AT-S-c C s BrO-abst-complex HOBr 1.756 BrO-AT-S-t C 1 BrO C O S Br C 2.462 1.83 120. CS 10.4 C O Br S C 2.472 1.80 1.3 C 10.5 C S O Br C 2.095 0.983 1.869 104.8 C 10.8 OBr C Br O S C C 10.5 2.456 17.2 1.82 O Br 1.871 103.2 0.968 BrO-SMe 2 -c C s C O Br S C 13.4 2.04 1.972 CS 10. C Br O S C 2.041.972 13. C 10. C S C Br O 1.2761.280 1.82 C 10.2 BrO-SMe 2 -t C 1 BrO-abst-TS C 1 85. 95. 95. C Br O S C 1.964 2.013 C 10.1 129.8 7.8 Figure 3. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p) level. Bond lengths are in ? and angles are in degres. in HBr (86.9 kcal?mol -1 ) is smaler than the calculated C-H bond enthalpy in DMSO (98.5 kcal?mol -1 ). 49 0. C C S DMS BrO OBr C O S Br C BrO-SMe 2 -t C 1 C Br O S C BrO-AT-S-c C s O Br HOBr C C S MeSCH 2 C S O Br C -1.7 -0.4 2.0 -8.7 -4.0 -30.5 C O S C + + + BrO-abst-complex BrO-abst-TS MeS(O)Me Br C S C Br O Figure 4. Schematic diagram of the potential energy surface computed at the G3B3(MP2) (298K) level for the reaction of BrO with DMS. Relative energies are given in kcal/mol at 298 K The abstraction transition state is 2.4 kcal?mol -1 higher in enthalpy than the OAT transition state, consistent with the experimental results that the abstraction pathway is 50 not observed. 6,8-10 The calculated rate constant 34 for the OAT and hydrogen abstraction pathways are 8.7x10 -13 cm 3 ?s -1 and 8.9x10 -15 cm 3 ?s -1 , respectively at 298K and 760 torr which gives a branching ratio of 98. The calculated rate constant for the OAT pathway (8.7x10 -13 cm 3 ?s -1 ) is in good agrement with experiment (3?1)x10 -13 cm 3 ?s -1 ). 2.3c IO + DMS: The trans IO-DMS complex (IO-SMe 2 -t C 1 ) is more stable than the cis complex (IO-SMe 2 -c C s ) by 1.2 kcal?mol -1 . The nonbonded I-H distance in the cis IO- DMS complex (3.356 ?) is larger than the sum of the vdW radii of iodine (2.16 ?) 3a nd hydrogen (1.10 ?) 3b which may indicate litle H-bonding. While the lowest orientation of the IO-DMS complex is trans, the cis OAT transition state (IO-OAT-TS-c) is 3.3 kcal?mol -1 lower in energy than the trans OAT (IO-OAT-TS-t C 1 ). Thus, the reaction path for the OAT reaction is predicted to start with IO in the trans orientation and reach the TS in the cis orientation which would suggest that the departing iodine radical should have significant angular motion. The enthalpy of the OAT transition state is lower than the enthalpy of the IO- DMS complex (-1.3 compared to ?2.6 kcal?mol -1 ) which is not possible on a continuous PES but does occur when enthalpies are derived from a composite method such as G2B3(MP2). It should be noted that the fre energy (298K) of the OAT transition state, IO-OAT-TS-c, is 6.9 kcal?mol -1 higher than the fre energy of IO + DMS, and the same fre energy as the complex, IO-SMe 2 -t C 1 (Table 3). Nakano et al. 10 have determined that the OAT reaction has a negative activation energy of ?4.4 kcal?mol -1 . The abstraction transition state is significantly higher in enthalpy than the OAT transition state (3.3 compared to ?2.6 kcal?mol -1 ). Since the bond enthalpy in the X-O 51 C O I S C OI 1.981 IO C O I S C 2.3782.03 123. C 10.5 C O I S C 2.06 2.410 128. C 10.4 IO-SMe 2 -t C 1 IO-SMe 2 -t C s IO-SMe 2 -c C s 2.06 2.147 13. C 10.9 C O I S C C 10.9 2.052.147 13.5 C S I C O 1.3 1.247 2.03 10.9 C 10.9 IO-abst-TS C 1 C S O C I 2.087 105.9 2.073 0.980 C 10.6 OI HOI 2.072 0.968 IO-AT-S-t C 1 IO-AT-S-t s IO-AT-S-c C s 105. 84.7 93.6 IO-abst-complex C 1 C I O S C 2.098 2.15 98. C 10.9 128. C I O S C 2.89 2.061 25.1 C 10.8 98.4 Figure 5. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d,p)/ECP level. Bond lengths are in ? and angles are in degres. bond should decrease in the order Cl-O > Br-O > I-O, it is expected that the OAT activation barier should decrease in the same order. This expectation is fulfiled for the 52 0. -1.3 3. -10.9 -5.8 -2.6 -38. OI C S C C I O S C C O S C C S C OI C SO C I C S C I O C S O C I + + DMS I IO-SMe 2 -t C 1 IO-abst-TS IO-AT-S-c C s IO-abst-complex HOI MeSCH 2 MeS(O)Me + I Figure 6. Schematic diagram of the potential energy surface computed at the G2B3(MP2) (298K) level for the reaction of IO with DMS. Relative energies are given in kcal/mol at 298 K. 53 OAT activation enthalpies (1.3 > -0.4 > -2.6 kcal?mol -1 for ClO, BrO, and IO, respectively). On the other hand, the abstraction of hydrogen occurs from the oxygen end of XO and is not expected to change significantly (2.1, 2.0, and 3.3 kcal?mol -1 for ClO, BrO, and IO, respectively). The OAT reaction enthalpy to form I + DMSO is calculated to be ?38.3 kcal?mol -1 which is significantly more exothermic than the experimental value 17 of ?29.0 kcal?mol -1 . It should be noted that the SOC of the I atom (which is included in the calculations) contributes 7.25 kcal?mol -1 to the exothermicity. The iodine radical is not expected to abstract a hydrogen from DMSO because the calculated bond enthalpy of HI (63.6 kcal?mol -1 ) is much smaler than the calculated H-C bond enthalpy in DMSO (98.5 kcal?mol -1 ). The abstraction pathway generates the HOI/MeSCH 2 complex which is bound by 5.1 kcal?mol -1 with respect to separated products. Table 4. Calculated Rate Constants (cm 3 ?s -1 ) and Branching Ratios of XO + DMS (X=Cl, Br, I) at 298K for Oxygen-Atom Transfer (OAT) and Hydrogen Abstraction Pathways OAT (k 1 ) Abstraction (k 2 ) Branching Ratio (k 1 /k 2 ) ClO+DMS 3.0x10 -15 1.1x10 -15 2.7 BrO+DS 8.7x10 -13 8.9x10 -15 98 IO+DMS 1.5x10 -1 1.0x10 -15 15000 The calculated rate constant 34 of the OAT pathway (1.5x10 -1 cm 3 ?s -1 ) is much faster than the abstraction pathway (1.0x10 -15 cm 3 ?s -1 ). While there is considerable disagrement among the experimental determinations of the rate constant (Table 1), the 54 present calculations support a fast reaction betwen IO and DMS. A summary of the calculated rate constants is given in Table 4. 2.3d Bonding in XO-DMS Aducts: An estimate of the binding enthalpy of an unsymmetrical 2c-3e complex is given by the average of the symmetrical complexes times e -?IE (eq 7), where ?IE is the diference (measured in eV) in ionization energies, a factor related to ?orbital matching?. 35 D AB =(D A +D BB )/2)?e -?IE (7) An adjustment must be made for the ionization energy of XO - to take into acount that a work term for separating charge is ?mising? (Table 5). If the PseudoIE of XO - is used rather than the EA (Table 5), then an estimate of orbital matching can be made. Using the PseudoIE from Table 5, the orbital match for HO-DMS (?IE=0.76=8.69-7.93, eV) is much beter than the orbital match for ClO-DMS (?IE=2.30=8.69-6.39, eV) which leads to a much greater predicted binding enthalpy for HO-DMS (eq 7, 14.8 kcal?mol -1 ) 36 compared to ClO-DMS (eq 7, 2.5 kcal?mol -1 ). The corresponding estimates of the BrO- DMS and IO-DMS binding enthalpies from eq 7 are also smal, 2.8 and 3.1 kcal?mol -1 , respectively. Thus, the ClO, BrO, and IO radicals are qualitatively diferent from OH in that the orbital match with DMS is poorer which leads to much smaler binding enthalpies. The calculated DFT binding enthalpies for XO-DMS (6.7, 6.1, and 7.8 kcal?mol -1 for ClO-DMS, BrO-DMS, and IO-DMS, respectively) are substantialy larger 55 than the G3B3, G3B3(MP2), and G2B3(MP2) values (Table 3) due to known deficiencies in the DFT method for describing theses systems. 37 The bonding betwen ClO and DMS in the ClO-DMS complex, as determined by an NBO population analysis at the B3LYP/6-311+G(d,p) level, 38 consists of one ? bond occupied by a single ?-spin electron. There are no corresponding ?-spin electrons in S-O bonding or antibonding orbitals, rather two unpaired ?-spin electrons reside in nonbonding (singly-occupied lone-pair) orbitals (Figure 7). The ?-bond is polarized toward sulfur (61.9%, Table 6). Since DMS donates 0.34 electrons to ClO, a most !* so LP " # LP " LP # LP " XO DMS! so Figure 7. Interaction diagram of the singly occupied molecular orbital (SOMO) of XO (X=Cl, Br, I) interacting with the highest occupied molecular orbital (HOMO) of DMS. consistent interpretation of the bonding in the ClO-DMS complex is the formation of a ?- electron dative bond (Figure 7), i.e. a ?-bond (ClO?SMe 2 ). The interpretation of the XO-DMS bonding for X=Br and I is very similar to ClO-DMS. 56 Table 5. Experimental Binding Enthalpies (kcal?mol -1 ) of X?X - , Calculated Binding Enthalpies (kcal?mol -1 ) of XO?OX - , and PseudoIE (eV) of X - and XO - A Binding Enthalpy a of A?A - exptl. EA (eV) of A Work term b (eV) PseudoIE (eV) of A - F 30.2 3.40 6.09 9.49 Cl 31.8 3.62 5.18 8.80 Br 27.9 3.36 4.52 7.88 I 24.3 3.06 4.12 7.18 OH 33.6 c 1.84 6.09 7.93 OCl 20.6 2.27 4.12 6.39 OBr 21.4 2.35 4.12 6.47 OI 26.5 2.37 4.12 6.49 a) The binding enthalpies of X?X - (X=F, Cl, Br, I) are exptl. values (se: Bra?da, B.; Hiberty, P. C. J. Phys. Chem. A 2000, 104, 4618 and Chermete, H.; Ciofini, I.; Mariotti, F.; Daul, C. J. Chem. Phys. 2001, 115, 11068). The binding enthalpies (298K) of XO?OX - (X=H, Cl, Br, I) are calculated at the B3LYP/6-311+G(2df,p) level of theory. The IO?OI - calculation used a basis set of similar size on Iodine with an ECP replacing the core electrons. The asumed symmetry of the XO?OX - complex was C 2 for X=H and Cl and C 2h for X=Br and I. The optimized O?O distances in the XO?OX - complexes were 2.419, 2.294, 3.517, and 3.428 ? for X=H, Cl, Br, and I, respectively. The calculated binding enthalpy (298K) of (DMS) 2 + at the B3LYP/6-31+G(d) level of theory is 29.8 kcal?mol -1 . Spin-orbital corrections are not included in the calculation of XO?OX - binding enthalpies. b) Work necesary to separate unit charges (work=qQ/4?? o R) where R is asigned a reasonable value (Wine, P. H.; McKe, M. L. to be published). This term is added to the EA of A to give the PseudoIE of the anion (A - ). A comparison of the PseudoIE of A - and the IE of DMS (exptl. 8.69 eV) wil give an indication of orbital matching. A beter orbital match betwen A - and DMS wil lead to a stronger 2c-3e interaction. c) The 2c-3e structure of HO?OH - is known to colapse to lower-energy structures. Se: Bra?da, B.; Thogersen, L.; Hiberty, P. C. J. Am. Chem. Soc. 2002, 124, 11781. 57 Table 6. NPA Charges, Total Atomic Spin Densities, and ?-bond Polarization of the ? so bond in the XO-DMS complex (X=Cl, Br, I) a NPA Charge Spin Densities ?-bond Polarization XO DMS O S O S ClO-SMe 2 -c C s CcccCCsC1 C s -0.34 0.34 0.54 0.37 38.1% 61.9% BrO-Se 2 -t 1 -0.36 0.36 0.54 0.38 39.7% 60.3% IO-SMe 2 -t C 1 -0.40 0.40 0.54 0.41 41.9% 58.1% a At the B3LYP/6-311+G(d,p) for X=Cl, and Br and at the B3LYP/6-311+G(d,p)/ECP level for X=I. 2.4 Conclusions The reaction rates betwen XO and DMS were computed for the oxygen-atom transfer (OAT) and hydrogen abstraction pathways. The relative reactivity of XO toward DMS is IO > BrO > ClO at 298K and 760 torr with rate constants of 3.0x10 -15 cm 3 ?s -1 , 8.7x10 -13 cm 3 ?s -1 and 1.5x10 -1 cm 3 ?s -1 for ClO, BrO and IO, respectively at 298K and 760 torr. The abstraction pathway is les favored with calculated branching ratios of 2.7, 98, and 15000 for ClO, BrO and IO, respectively. The calculated rate constants for ClO and BrO with DMS are in good agrement with experiment, while the calculated rate constant for the IO plus DMS reaction overlaps at the ?fast? end of experimental results. The binding enthalpies for ClO-DMS, BrO-DMS and IO-DMS complexes (2.0, 1.7, and 1.3 kcal?mol -1 , respectively) are much weaker than the HO-DMS complex (13?3 kcal?mol -1 ) due to a poorer orbital match betwen XO and DMS (compared to OH and DMS). 58 2.5 References 1. Charlson, R. J.; Lovelock, J. E.; Andreae, M. O.; Waren, S. G. Nature 1987, 326, 655. 2. Barone, S. B.; Turnipsed, A. A.; Ravishankara, A. R. J. Chem. Soc. Faraday Trans. 1995, 100, 39. 3. Tyndal, G.; Ravishankara, A. R. Int. J. Chem. Kinet. 1991, 23, 483. 4. Yin, F.; Grosjean, D.; Seinfeld, J. H. J. Atmos. Chem. 1990, 11, 309. 5. Barnes, I.; Becker, K. H.; Martin, D.; Carlier, P.; Mouvier, G.; Jourdan, J. L.; Laverdet, G.; Le Bras, G. ?Impact of Halogen Oxides on Dimethyl Sulfide Oxidation in the Marine Atmosphere?, In Biogenic Sulfur in the Enviroment, E. S. Saltzman, E. S, Cooper, W. J., Eds, ACS Symposium Series, 1989, 393, 464. 6. Barnes, I.; Bastian, V.; Becker, K. H.; Overath, R. D. Int. J. Chem. Kinet. 1991, 23, 579. 7. D?az-de-Mera, Y.; Aranda, A.; Rodr?guez, D.; L?pez, R.; Caba?as, B.; Mart?nez, E. J. Phys. Chem. A 2002, 106, 8627. 8. Bedjanian, Y.; Poulet, G.; Le Bras, G. Int. J. Chem. Kinet. 1996, 28, 383. 9. Ingham, T.; Bauer, D.; Sander, R.; Crutzen, P.J.; Crowley, J. N. J. Phys. Chem. A. 1999, 103, 7199. 10. Nakano, Y.; Goto, M.; Hashimoto, S.; Kawasaki, M.; Walington, T. J. J. Phys. Chem. A 2001, 105, 11045. 11. Barnes, I.; Becker, H. K.; Carlier, P.; Mouvier, G. Int. J. Chem. Kinet. 1987, 19, 489. 59 12. Martin, D.; Jourdain, J. L.; Laverdet, G.; Le Bras, G. Int. J. Chem. Kinet. 1987, 19, 503. 13. Daykin, P. E.; Wine, P. H. J. Geophys. Res. 1990, 95, 18547. 14. Maguin, F.; Melouki, A.; Laverdet, G.; Poulet, G.; Le Bras, G. Int. J. Chem. Kinet. 1991, 23, 237. 15. Knight, G. P; Crowley, J. N. Phys. Chem. Chem. Phys. 2001, 3, 393. 16. Nakano, Y.; Enami, S.; Nakamichi, S.; Aloisio, S.; Hashimoto, S.; Kawasaki, M. J. Phys. Chem. A 2003, 107, 6381. 17. The webbook (http:/webbook.nist.gov/chemistry) was used as the source of al thermochemistry except for the heat of formation of BrO and IO. (a) BrO: Bedjanian, Y.; Le Bras, G.; Poulet, G. Chem. Phys. Let. 1997, 266, 233. (b) IO: Bedjanian, Y.; Le Bras, G.; Poulet, G. J. Phys. Chem. 1996, 100, 15130. 18. Gaussian 98 (Revison A11), Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Milam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cami, R.; Mennucci, B.; Pomeli, C.; Adamo, C.; Cliford, S.; Ochterski, J.; Peterson, G. A.; Ayala, P. Y.; Cui, Q.; Morukuma, K.; Salvador, P.; Dannenberg, J. J.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Chalacombe, M.; Gil, P. M. 60 W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, C.; Head- Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian, Inc., Pitsburgh PA, 2001. 19. MOLDEN, Schaftenaar, G.; Noordik, J. H. J. Comput.-Aided Mol. Design 2000, 14, 123. 20. Curtis, L. A.; Raghavachari, K.; Redfern, P. C; Rasolov, V.; Pople, J. A. J. Chem. Phys. 1998, 109, 7764. 21. Curtis, L. A.; Raghavachari, K.; Pople, J. A. J. Chem Phys. 1993, 98, 1293. 22. Glukhovtsev, M. N.; Pross, A.; McGrath, M. P.; Radom, L. J. Chem. Phys. 1995, 103, 1878. Eratum: 1996, 104, 3407. 23. Curtis, L. A.; Redfern, P. C.; Raghavachari, K.; Rasolov, V.; Pople, J. A. J. Chem. Phys. 1999, 110, 4703. 24. Curtis, L. A.; Redfern, P. C.; Rasolov, V.; Kedziora, G.; Pople, J. A. J. Chem. Phys. 2001, 114, 9287. 25. (a) Schwerdtfeger, P.; Dolg, M.; Schwarz, W. H.; Bowmaker, G. A.; Boyd, P. D. W. J. Chem. Phys. 1989, 91, 1762. (b) Bergner, A.; Dolg, M.; K?chle, W.; Stoll, H.; Preuss, H. Mol. Phys. 1993, 80, 1431. 26. Furlani, T. R.; King, H. F. J. Chem. Phys. 1985, 82, 5577. King, H. F.; Furlani, T. R. J. Comput. Chem. 1988, 9, 771. Fedorov, D. G.; Gordon M. S. J. Chem. Phys. 2000, 112, 5611. 27. Stevens, J. E.; Cui, Q.; Morokuma, K. J. Chem. Phys. 1998, 108, 1554. 28. Mokrushin, V.; Tsang, W. CHEMRATE. A Calculational Data Base for Unimolecular Reactions; National Institute of Standards and Technology: Gaithersburg, MD, 2000. 61 29. (a) NBO Version 3.1, Glendening, E. D.; Red, A. E.; Caroenter, J. E.; Weinhold, F. (b) Re d, A. E.; Curtis L. A.; Weinhold, F. Chem. Rev. 1988, 88, 899. 30. Vandresen, S.; Resende, S. M. J. Phys. Chem. A 2004, 108, 2248. 31. Stickel, R. E.: Nicovich, J. M.; Wang, S.; Zhao, Z.; Wine, P. H. J. Phys. Chem. 1992, 96, 9875. 32. Butkovskaya, N. I.; Poulet, G.; Le Bras, G. J. Phys. Chem. 1995, 99, 4536. 33. (a) Batsanov, S. S. J. Chem. Soc. Dalton Trans. 1998, 1541. (b) Mandal, P. K.; Arunan, E. J. Chem. Phys. 2001, 114, 3880. 34. Due to a limitation of the CHEMRATE program, we could not use a negative activation barier. Therefore, we asumed an activation barier as zero and multiplied the rate by e -?H/RT where ?H is the activation enthalpy (i.e ?H a =-0.4 or ?H a =-2.6 kcal?mol -1 for the OAT reaction of BrO and IO with DMS, respectively). 35. Clark, T. J. Am. Chem. Soc. 1988, 110, 1672. 36. The experimental binding enthalpy is 13.0 kcal?mol -1 : (a) Hynes, A. J.; Stoker, R. B.; Pounds, A. J.; McKay, T.; Bradshaw, J. D.; Nicovich, J. M.; Wine, P. H. J. Phys. Chem. 1995, 99, 16967. The calculated binding entrhalpy is 9-10 kcal?mol - 1 : (b) Turec?ek, F. Collect. Czech. Chem. Commun. 2000, 65, 455. (c) Wang, L.; Zhang, J. THEOCHEM 2001, 543, 167. (d) McKe, M. L. J. Phys. Chem. A 2003, 107, 6819. 37. For a summary of deficiencies of the DFT methods for describing 2c-3e systems se: (a) Gr?fenstein, J.; Kraka, E.; Cremer, D. Phys. Chem. Chem. Phys. 2004, 6, 1096. (b) Fourr?, I.; Berg?s, J. J. Phys. Chem. A 2004, 108, 898. 62 38. A NBO analysis at the MP2/6-311G(d,p) level (with the MP2 density matrix, i.e. DENSITY=CURENT) gives a very similar description of the bonding for the ClO-DMS complex. 63 CHAPTER 3 THEORETICAL STUDY OF THE MECHANISM OF NO 2 PRODUCTION FROM NO + ClO 3.1 Introduction The concentrations and chemistry of the ClO x and NO x radicals are important for understanding the global atmospheric chemistry. 1 The ClO x and NO x radicals are involved in tropospheric ozone production and stratospheric ozone loss. 2,3 Generaly, chlorine (Cl) is oxidized by ozone in the stratosphere and forms ClO which can be removed by other reactions. Nitrogen oxides (NO x ) act as sinks for ClO, which are transformed into temporary reservoir species, such as ClONO 2 and HCl. These reservoir species do not react with ozone and are slowly removed from the stratosphere. Although there have been many studies on atmospheric chlorine chemistry, there are stil discrepancies concerning the atmospheric chlorine budget. 4,5 Specificaly, there is a mising reservoir of inorganic chlorine in the stratosphere which may not be acounted for by ClONO 2 and HCl alone. For example, it is possible that nitryl chloride (ClNO 2 ) could also be an important chlorine reservoir. 6 Photolysis of ClNO 2 is predicted to be rapid in sunlight and may be the dominant los mechanism, yielding primarily atomic chlorine. 64 The recombination of ClO and NO radicals is known to produce NO 2 and Cl radicals 8-13 (eq 1-4) through possible involvement of ONOCl and/or ClNO 2 as intermediates. ClO + NO ? ONOCl (1) ONOCl ? Cl + NO 2 (2) ONOCl ? ClNO 2 (3) ClNO 2 ? Cl + NO 2 (4) Experimental studies on the NO and ClO reaction betwen 200 and 400K, 9,10,13 have shown the reaction to have a negative activation barier. While the presure dependence of the reaction has not been reported, the concentration of the bath gas in the Leu et al. study 10 was much higher than the concentration of the reactants which may suggest that their results may be near the high-presure limit. Several studies 8,1,12 report rate constants at only one temperature (298K), and these results are in agrement with the temperature-dependence experimental results. 9,10,13 The ONOX (X=OH, F, Cl) potential energy surfaces are expected to share some similarity. 14-18 When X=OH, the concerted formation of HNO 3 from HONO is stil very much in doubt. 14,17,18 Zhao et al. 14 showed that there is no direct isomerization betwen HONO and HONO 2 . They explained the mechanism of O-O cleavage from cis-HONO to the formation of NO 2 + OH and calculated a 18-19 kcal/mol activation barier for this proces. On the other hand when X=F, Elison et al. 15 showed that ONOF forms FNO 2 on a continuously connected PES through a very loose transition state with an activation barier of 22?3 kcal/mol. Zhu and Lin 16 recently calculated an 65 isomerization path betwen cis-ONOCl and ClNO 2 with an activation barier of 21.2 kcal/mol (through NO 2 + Cl). In the following study, we wil use electronic-structure methods to calculate the potential energy surface for the formation of NO 2 + Cl from NO + ClO. We wil calculate the rate constant for disappearance of reactants (k dis ) as wel as the rate constant for formation of products (k obs ) over the temperature range 200-1000K. At the temperatures and presures used in the experimental studies, it is not known whether the dependence of the rate constant on temperature is in the fal-off or presure-independent regime. Since the concentration of bath gas is much larger than the concentration of reactants, the reaction may be close to presure independent. Al the calculated rate constants reported below are at the high-presure limit. 3.2 Computational Method Since B3LYP density functional theory (DFT) has been shown to give reasonable structures and vibrational frequencies for halogen compounds, 19-2 we decided to use that method to calculate the PES. However, to check the DFT results, we also decided to use a more acurate method to determine the minima, transition states, and thermodynamic properties. Since the CSD(T) method (with a reasonable basis set) yields very good results for dificult chemical systems such as O 3 and FOF, 23,24 we used it (with finite- diference derivatives) to optimize al of the stationary points on the PES. Al electronic structure calculations have used the Gaussian03 25 and Molcas6 26 program systems. Optimization and frequency calculations for the NO + ClO potential 66 energy surface were caried out at the B3LYP/6-311+G(d) and CSD(T) levels. Al imaginary frequencies for transition states were animated by using the graphical program MolDen 27 to make sure that the motion was appropriate for converting reactants to products. In transition states involving bond formation or bond breaking, a lower-energy spin broken symmetry solution was obtained at the UDFT or UHF levels. For DFT calculations, we used the spin-broken symmetry results. However, for the CSD(T) calculations, it was not clear whether the spin-restricted (RCSD(T) or spin-unrestricted (UCSD(T) method would produce more reliable results. In the work by Elison et al. 15 on the ONOF ? FNO 2 transition state a variety of post-SCF methods were used. At the RCSD(T)/DZP and RCSD(T)/pVTZ levels, the activation energies were 3 and 4 kcal/mol lower than at the UCSD(T)/DZP and UCSD(T)/pVTZ levels, respectively and in beter agrement with their best computational results. Thus, while the UHF reference state is lower in energy than the RHF for these transition states, it appears that the RHF reference may be a beter choice as the reference function for the post-SCF perturbative expansion. Our RCSD(T) and UCSD(T) results for the NO + ClO reactions exhibit the same behavior, but more extreme. With respect to separated NO + ClO radicals computed at the UCSD(T) level, the activation bariers for cis and trans addition were 5.36 and 7.49 kcal/mol, respectively. This result is in contrast to the B3LYP/6-311+G(d) results, where no transition state could be located, as wel as CASPT2 results on fixed ON-OCl geometries, where the energy decreased monotonicaly as the N-O distance decreased. The RCSD(T) method located transition states for cis and trans addition 67 which were 4.31 and 3.49 kcal/mol below NO + ClO. Thus, the RCSD(T) potential energy surface has a maximum for the formation of the N-O bond, but the stability of the entire surface at these large N-O distances is overestimated since it is below the energy of the reactants (NO + ClO). Table 1. Relative Energies a (kcal/mol) at the B3LYP/6-311+G(d), G3B3, CSD(T)/c-pVTZ/CSD(T)/c-pVDZ Levels for Various Species Involved in the NO + ClO Reaction ?H(0K) ?H(298K) ?G(298K) B3LYP G3B3 CSD(T)/ c-pVTZ B3LYP G3B3 CSD(T)/ c-pVTZ b B3LYP G3B3 CSD(T)/ c-PVTZ NO+ClO 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 t-ONOCl -27.08 -31.10 -27.37 -27.94 -31.94 -28.21(-28.3) -18.03 -2.02 -18.28 c-ONOCl -31.07 -32.78 -30.34 -32.06 -3.75 -31.34(-31.3) -21.9 -23.68 -21.15 ONOCl-ts -16.71 -21.9 -18.31 -17.85 -23.12 -19.46(-19.3) -7.61 -12.8 -9.12 ON-OCl-ts-c -3.86 -4.31 3.94 ON-OCl-ts-t -3.17 -3.49 3.76 ON-OCl-ts-c c 5.94 5.36 14.0 ON-OCl-ts-t c 8.10 7.49 16.43 NO 2 Cl-ts -18.84 -21.6 -17.20 -20.16 -2.96 -18.45(-18.0) -9.6 -12.46 -7.64 Abst-Cl-ts-t -3.8 10.71 4.54 -4.8 9.71 3.62 5.23 19.82 13.56 Abst-Cl-ts-c -10.75 -1.59 -2.93 Cl-ad-NO 2 -ts -16.29 -12.17 -10.17 -16.92 -12.7 -1.01 -1.46 -2.53 -1.85 Cl-ad-NO 2 -ts c -13.62 -15.7 -6.60 NO 2 + Cl -15.49 -10.12 -10.14 -15.79 -10.40 -10.42(-10.1) -13.79 -8.41 -8.86 ClNO 2 -46.01 -4.85 -41.04 -47.34 -46.13 -42.30(-42.2) -36.34 -35.13 -31.32 a The Spin-Orbit Correction (SOC) of NO (Ref. 31), ClO (Ref. 32), and Cl (Ref. 33) are included in G3B3 and CSD(T)/c-pVTZ level. b The values in parentheses are taken from Ref. 16. c Geometries are optimized at the UCSD(T)/c-pVDZ level. The CSD(T) optimizations and frequency calculations were caried out with the correlation-consistent c-pVDZ basis of Dunning 28 with single-point calculations using 68 the c-pVTZ basis set 29 (i.e. CSD(T)/c-pVTZ/CSD(T)/c-pVDZ). The G3B3 method 30 was used with manual asembly of the components where we used B3LYP/6- 311+G(d) optimized geometries and unscaled vibrational frequencies with al other calculations and corrections at standard levels. For transition states which were lower in energy using the UDFT spin broken-symmetry approach, the coresponding G3B3 calculations (including the post-SCF steps) were based on the spin-broken symmetry UHF wave function. The first-order spin-orbit corrections are included for NO, 31 ClO, 32 and Cl 3 (0.11, 0.30, and 0.78 kcal/mol, respectively) in al of the relative energy comparisons in Table 1. 34 The reaction coordinate for cis and trans approach of NO and ClO was constructed by optimizing the structure with spin broken-symmetry UDFT while fixing the N?O distance betwen the two radicals. In this way a set of structures, where the N?O distance was fixed to 4.00, 3.75, 3.50, 3.25, and 3.00 ?, was obtained. Harmonic vibrational frequencies were computed for each structure along the cis and trans reaction coordinate. At each structure, one imaginary frequency was obtained corresponding to the N?O stretch. The only exception was at a 4.00 ? separation along the trans reaction coordinate which was not used in the subsequent rate calculations. The large diference betwen the UCSD(T) and RCSD(T) energies for the transition states of the NO+ClO addition reactions indicates that electron correlation methods based on a single reference are not appropriate for this part of the potential energy surface. For that reason, single-point calculations were made with a 12-electron 12-orbital complete active space 35 (CAS(12,12)) and the ANO-L basis set 36 with dynamic electron correlation introduced at the MP2 level (CASPT2). 37 At each structure, four 69 singlet electronic states were computed, two 1 A? states and two 1 A? states. Both radicals, NO and ClO, have 2 ? ground states. As the two radicals approach, the electronic and spatial degeneracy wil be lifted to give rise to a total of sixten microstates, four singlet states and four triplet states. The ?reactive? state wil correspond to 1 A? where the unpaired electron on each radical approaches in the reaction plane. The other electronic states wil be occupied acording to the Boltzmann distribution. We computed the electronic partition function to determine the fraction of reactive encounters which are in the ?reactive? electronic state. At a short N?O separation where the ?reactive? electronic state is already stabilized, the fraction is 0.98 at 200K and 0.54 at 1000K (3.00 ?, cis addition). At 4.00 ? the fraction is 0.36 at 200K and 0.27 at 1000K (cis addition). Since we are only considering the four close-lying singlet states, complete degeneracy would result in a 0.25 factor for each state. Rate constants for the NO + ClO ? NO 2 + Cl reaction were calculated with Polyrate-9.3 38 and VariFlex-1.0. 39 Rate constants were computed at each N-O separation (4.00, 3.75, 3.50, 3.25, and 3.00 ?) as a function of temperature (Table S4). For the cis approach, the smalest rate constant was found at 4.0 ? at each temperature. Technical dificulties prevented us from calculating rate constants at longer N-O separations. For the trans approach, the smalest rate constant was obtained at a N-O separation of 3.50 ?. Since the formation of NO + ClO occurs without a barier on the potential energy surface, we used VariFlex, a program designed for this task, to calculate the rate constants for cis and trans addition. VariFlex is a variational RKM code that solves the master equation involving multistep vibrational energy transfer for the excited intermediate ONOCl. The enthalpy barier for trans ? cis isomerization of ONOCl is 70 8.75 kcal/mol (11.88 kcal/mol reverse barier ). We calculated rate constants with ChemRate 40 with harmonic frequencies at B3LYP/6-311+G(d) and energies at CSD(T)/c-pVTZ/CSD(T)/c-pVDZ. For calculations using Polyrate, we used B3LYP/6-311+G(d) for the IRC and harmonic frequencies and G3B3//B3LYP/6- 311+G(d) for energies. Cl O ON 1.76 .1 a 1.68 .1 a 1.49 .8 a 1.359 .40 a 16.9 5. a 19.3 7.5 a 15.2 3.6 a 18.2 6. a 15.7 b 1.74 .20 a 1.793 .4 a 1.732 b c-ONCl N O Cl O 1.65 . a 1.48 .6 a 1.529 .4 a 1.527 .3 a 107.9 .5 a 10.9 .8 a 1.741 .3 a 1.74 .02 a 107.4 8.1 a 10. 8.2 a O N O Cl 1.97 .8 a 1.7 .86 a 1.20 c 1.93 .87 a 1.94 .05 a 1.84 c 13.4 2.0 a 13.7 2.8 a 130.6 c ClNO 2 t-ONCl Figure 1. Optimized geometry of cis-, trans-ONOCl and ClNO 2 isomers. Bond lengths are in ? and angles are in degres. Data in the first row and third row are calculated at the CSD(T)/c-pVDZ and B3LYP/6-311+G(d) level, respectively. (a) Calculated values at the CSD(T)/TZ2P and B3LYP/TZ2P level, respectively are taken from Ref. 19. (b) and (c) are experimental values, are taken from Ref. 42 and Ref. 52, respectively. 71 Cl O O N -28.1 -31.4 -19.46 -18.45 3.62 -10.42 -42.30 -31.4 Cl O N O N O Cl O Cl O ON O N O Cl O N O Cl O O N + N O OCl Cl t-ONCl c-ONCl c-NCl ClNO 2 Abst-Cl-ts-t Cl-ts ClN 2 0. N 2 + Cl Figure 3. Schematic diagram of the potential energy surface for the NO + ClO system computed at the CSD(T)/c-pVTZ/CSD(T)/c-pVDZ level. Relative energies are given in kcal/mol at 298K. 3.3 Results and Discussion It is widely acepted that the reaction NO + ClO ? NO 2 + Cl involves the ONOCl intermediate which has two distinct conformers cis- and trans-ONOCl as wel as the ClNO 2 isomer for which there exists reliable experimental data. However, there is litle reliable experimental data available for the other ONOCl compounds. Since there is 72 no evidence for OClNO, 41 we did not include this isomer in our study. The equilibrium structure for cis-ONOCl (Figure 1) is consistent with the earlier studies and experimental results. 19,42 Figure 1 shows that the Cl-N bond distance of ClNO 2 has the largest sensitivity with respect to the method with CSD(T)/c-pVDZ giving the longest Cl-N bond distance and CSD(T)/TZ2P the shortest. Experimentaly, 43 the enthalpy of reaction of eq 3 is -42.87 kcal/mol which is almost same as the CSD(T)/c-pVTZ/CSD(T)/c- pVDZ result of -42.30 kcal/mol. The experimental 43 enthalpy diference betwen cis- ONOCl and reactants (NO + ClO) is 32.17 kcal/mol which is excelent agrement with CSD(T)/c-pVTZ/CSD(T)/c-pVDZ (31.34 kcal/mol Figure 2) . A comparison of calculated and measured frequencies of the cis- and trans- ONOCl and ClNO 2 is presented in Table 2. The calculated frequencies are in agrement with each other for cis-ONOCl with slightly les agrement for the torsion (? 6 (a"). For ClNO 2 , the calculated frequencies agred with each other and are close to the experimental results. The NO + ClO system studied here is similar to the reaction of XO (X=H, F, Cl) and alkylperoxy radicals (RO) with NO. 14,4 A negative activation energy is observed for the NO + ClO reaction, which strongly implies the involvement of an ONOCl intermediate. The observed negative activation enthalpy is consistent with the negative activation bariers calculated for the trans and cis transition states (ON-OCl-ts-t and ON- OCl-ts-c, respectively) at the RCSD(T)/c-pVTZ/RCSD(T)/c-pVDZ level (Table 1, -3.49 and ?4.31 kcal/mol, respectively). 73 Table 2. Harmonic frequencies of trans-ONOCl, cis-ONOCl and ClNO 2 in cm -1 Method t-ONOCl ? 1 (a?) ? 2 (a?) ? 3 (a?) ? 4 (a?) ? 5 (a?) ? 6 (a?) B3LYP/TZ2P a 1834 879 660 404 259 178 B3LYP/6-311+G(d) 1854 874 645 403 255 182 CSD(T)/TZ2P a 1754 855 607 407 212 170 CSD(T)/c-pVDZ 1800 852 646 403 263 173 c-ONOCl ? 1 (a?) ? 2 (a?) ? 3 (a?) ? 4 (a?) ? 5 (a?) ? 6 (a?) B3LYP/TZ2P a 1741 868 647 365 229 398 B3LYP/6-311+G(d) 1732 868 673 363 212 426 CSD(T)/TZ2P a 1715 850 638 416 249 341 CSD(T)/c-pVDZ 1731 859 618 430 245 378 Expt b 1715 858 644 406 260-280 344 ClNO 2 ? 1 (a 1 ) ? 2 (a 1 ) ? 3 (a 1 ) ? 4 (b 1 ) ? 5 (b 2 ) ? 6 (b 2 ) B3LYP/TZ2P a 1339 810 370 673 1748 409 B3LYP/6-311+G(d) 1350 809 364 668 1774 411 CSD(T)/TZ2P a 1290 805 371 658 1688 409 CSD(T)/c-pVDZ 1342 798 345 640 1801 390 Expt c 1286 793 370 652 1685 408 a Ref. 19. b Janowski, B.; Knauth, H. D.; Martin, H. Ber. Bunsunges. Phys. Chem. 1977, 81, 1262. c Shimanouchi, T. J. Phys. Chem. Ref. Data 1977, 6, 993. Computational studies to find a low activation barier for the RONO and HONO unimolecular isomerization have failed. 14,18,4 In the study by Zhao et al., 14 the transition state for cis-HONO to NO 2 + OH was located 3 kcal/mol above NO 2 + OH radicals at the UCSD(T)/6-31+G(d)/UCSD/6-31+G(d) level. In our system, the transition state for cis-ONOCl to NO 2 + Cl is found at the RCSD(T)/c-pVDZ level. The stationary point has one imaginary frequency and is 1.17 kcal/mol lower than NO 2 + Cl at the RCSD(T)/c-pVTZ/RCSD(T)/c-pVDZ level. The trans-ONOCl ? ClNO 2 74 O Cl O N OCl N O O O N Cl O N O O N O Cl Cl O O N Cl O O N O Cl N O 1.60 1.743 1.52 10. 1.268 12. 1.2 2.19 108. 2.93 1.203 135.6 1.20 1.20 127. 2.53 3.189 135. 1.20 1.649 1.207 134. 1.6 Cl NO 2 NOCl-ts Abst-Cl-ts Abst-Cl-tsc Cl-ad-NO 2 -ts NO 2 Cl-ts 1.5 1.698 13.5 104.2 2.48 Cl O ON 10. 107.9 1.52 2.50 1.697 ON-Cl-tsc ON-Cl-tst (.48) (.7) (.93) (.) (.5) (.3) (.8) (.) (.067) (.2) (.7) (.93) (.) (.83) (.4) (.47) (5.) (.93) (4.) [.78] [.] [8.1] [8.4] [.7] [.][9.] [.693] [09.4] [.4] [2.936] [1.20] [13.2] Figure 2. Optimized geometric parameters of stationary points at the CSD(T)/c- pVDZ level with B3LYP/6-311+G(d) values in parentheses. Bond lengths are in ? and angles are in degres. The geometric parameters for ON-OCl- ts-c and ON-OCl-ts-t are at the RCSD(T)/c-pVDZ level with values at the UCSD(T)/c-pVDZ level given in brackets. 75 reaction has a much higher activation barier than the cis isomer. The N?Cl distance in the transition state (Figure 3, abst-Cl-ts-t) is much shorter (2.191 ?) than the N?Cl distance in abst-Cl-ts-c (2.939 ?). In addition, the two N-O distances are much more asymmetric in abst-Cl-ts-t compared to abst-Cl-ts-c. Basicaly, the abst-Cl-ts-c and abst-Cl-ts-t transition states are indistinguishable from fragmentation into NO 2 + Cl radicals. In their study of the c-HONO ? HONO 2 reaction, Dixon et al. 18 located a transition state with an activation energy of 21.4 kcal/mol using MP2/c-pVTZ where the transition state was 1.6 kcal/mol higher than NO 2 + OH. Zhao et al. 14 located a similar transition state at the CBS-QB3 (Complete Basis Set) level 21.0 higher than c-HONO and 2.7 kcal/mol higher than a NO 2 /OH complex, but they were unable to say whether the transition state corresponded to fragmentation or isomerization. The N?O distance in the two studies were quite long (2.784 and 3.070 ? at MP2/c-pVTZ and CBS-QB3, respectively). In contrast to the c- HONO ? HONO 2 reaction, the t-ONOF ? FNO 2 and t-ONOCl ? ClNO 2 reactions have much tighter transition states. Elison el al. 15 calculated a short breaking O?F distance in the t-ONOF ? FNO 2 transition state (1.726 and 1.693 ? at RCSD(T)/pVTZ and UCSD(T)/pVTZ, respectively) while we calculate a 2.191 ? N?Cl distance at RCSD(T)/c-pVDZ. The energy diference betwen abst-Cl-ts-t and abst-Cl-ts-c is 15.21 kcal/mol at the RCSD(T)/c-pVTZ/RCSD(T)/c-pVDZ level. The reason for the large activation energy diference can be explained with a similar explanation as given in the study of O- O cleavage in ONONO 45 and the mechanism of peroxynitrous acid and methyl 76 peroxynitrite (RONO). 14 The singly-occupied a 1 and doubly occupied b 2 orbitals of NO 2 fragment can mix through the lowering of symmetry from C 2v to C S caused by the OO N plus/minus phase cobiatio a 1 b 2 C 2v to C s orbital mixng O N O O N O a' a' Cis Aproach -Orbital Mixng Mechanism Trans Aproach -Electron Promtion Mechanism OO N a 1 b 2promtion but litle ixg rquird a' a' OO N Cl Cl a) b) Figure 4. Ilustration of cis and trans chlorine addition to NO 2 to form (a) cis-ONOCl and (b) trans-ONOCl. Cis addition can be rationalized by an orbital mixing mechanism. Trans addition has a higher activation barier and involves an electron promotion mechanism. 77 approaching Cl radical (Figure 4a). The mixing causes unpaired spin density to reside on the oxygen atoms. The oxygen atom that wil form a bond with chlorine has the larger lobe pointing toward the interior angle of the NO 2 fragment. A beter overlap betwen this lobe and the chlorine atom results when the chlorine atom approaches from the cis side compared to the trans side. The mechanism for N?Cl bond formation in NO 2 + Cl ? trans-ONOCl (Figure 4b) can be viewed as initial electronic promotion followed by bond formation. The electronic reorganization involves the promotion of a ?-spin electron from a b 2 orbital to an a 1 orbital which leaves an unpaired electron on oxygen in an orbital with significant extent away from the interior angle and suitable for bond formation with a chlorine atom approaching from that direction. Thus, bond formation from the ?cis side? of NO 2 requires orbital mixing while bond formation from the ?trans side? of NO 2 requires electronic promotion. The need for electronic promotion is source of the greater energy of abst-Cl-ts-t compared to abst-Cl-ts-c. An alternative explanation in term of correlating the reaction path with the 2 A 1 or 2 B 2 electronic state of NO 2 is also possible. 14,45 3.4 Rate calculations Radical-radical recombinations have long presented experimental and theoretical dificulties. 46,47,48 The very fast rates require specialized experimental techniques while reactions with no activation bariers are dificult to model theoreticaly. The phase- space-integral based VTST (PSI-VTST) method, as implemented in VariFlex, was used to evaluate the reactive flux as the N-O distances increased from 1.6 to 4.0 ? with a step size 0.1 ? for cis-ONOCl and trans-ONOCl intermediates. In order to evaluate the 78 reactive flux acurately, the transition from fre rotation to hindered rotation must be treated correctly. We also used variational transition state theory, as implemented in Polyrate, to calculate the rate constant for disappearance (k dis ) of reactants (NO + ClO). The reactive flux is calculated at a set of structures optimized at the UB3LYP/6-311+G(d) level with fixed central N-O distances of 3.0, 3.25, 3.50, 3.75, and 4.00 ?. The minimum reactive flux was found at 3.50 ? for trans-ONOCl. For cis-ONOCl the smalest flux was at 4.00 ? which we used as the minimum value because we could not compute the rate constant at larger N?O separations. We also considered the electronic partition function which was explained in detail in study of the O( 3 P) + OH reaction by Graf and Wagner. 49 The factor was calculated from eq 5 where p(T) is the ! p T ( ) k 0 T ( ) = 1 e "# j kT j $ k 0 T ( ) (5) probability that the collision involves the ?reactive? electronic state and k 0 (T) and ? j are the rate constant and the electronic energy (relative to lowest energy) summed over j states. Four singlet states were calculated (j=4, two 1 A? and two 1 A?) at the CASPT2(12,12)/ANO-L level using B3LYP/6-311+G(d) structures at fixed N?O separations. The rate constant for disappearance of reactants (k dis ) is the sum of k 1 and k 3 from the mechanism in Scheme 1. The rate constant for formation of products k obs is 79 NO + ClO c-ONCl Cl + NO 2 t-ONCl k -1 k 3 k 2 k 1 k -4 k 4 k -3 Scheme 1 derived with a steady state approximation for cis- and trans-ONOCl ( ! k obs = k 1 k 2 k "3 + k 4 ( ) + k 2 k 3 k 4 k "3 + k 4 ( ) k "1 + k 2 ( ) + k "3 k "4 ). The barier for direct formation of NO 2 + Cl (31.83 kcal/mol) from trans-ONOCl is much higher than the barier for isomerization to cis- ONOCl (8.75 kcal/mol). Thus, our mechanism includes direct formation of NO 2 + Cl from cis-ONOCl, but isomerization then fragmentation of cis-ONOCl for trans-ONOCl. Figure 5 compares the calculated VariFlex and Polyrate rate constants (high- presure limit) with experiments and the calculations of Zhu and Lin. 16 Our rate constants are in good agrement with experiment over the temperature range of 200- 400K. Over the temperature range 200-1000K, we fit our VariFlex rate constant data (k obs ) to the form 7.38x10 -13 T 0.413 exp(286/T) cm 3 ?molecule -1 ?s -1 . In comparison, the VariFlex rate constant for k dis (k 1 + k 3 ) can be fit to the form k dis =3.30x10 -13 T 0.58 exp(305/T) cm 3 ?molecule -1 ?s -1 . Before computing the Polyrate rate constants k dis , the CASPT2 energies (relative to the NO + ClO asymptote) were raised by 1.0 kcal/mol in order achieve beter agrement with experiment. 80 7 10 -12 8 10 -12 9 10 -12 10 -11 2 10 -11 1 1.5 2 2.5 3 3.5 4 4.5 5 k dis =k cis +k trans (Polyrate) k obs (VariFlex) k dis =k cis +k trans (VariFlex) k cis (VariFlex) k trans (VariFlex) Zho and Lin (ref. 16) ref. 9 ref. 10 ref. 13 R a t e C o n s t a n t k / c m 3 molecule -1 s -1 1000/T K -1 Figure 5. Calculated and experimental rate constants for the ClO + NO reaction. Al references are experimental rate data except Ref. 16 which is computational. The computational rate data are for the bimolecular rate constant at the high-presure limit. The thin lines for the cis and trans rate constants are added together to give k dis . 81 400 800 1200 1600 2000 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ! (N=O) " (ONO) ! (N-O) ! (Cl-O) " (ClON) V i b r a t i o n a l F r e q u e n c i e s ( c m - 1 ) s (a.u.) Figure 6. Plot of vibrational frequencies (cm -1 ) along the IRC for trans ? cis isomerization with the reaction projected out. The level of theory is B3LYP/6-311+G(d). The downward (convex) curve of the Zhu and Lin plot reveals non-Arhenius behavior at high temperature and is due to the negative exponent of temperature in their expresion k obs =1.43x10 -9 T -0.83 exp(92/T) cm 3 ?molecule -1 ?s -1 . Our plot indicates an upward (concave) curve due to the positive exponent of temperature in our rate constant expresion k obs =7.38x10 -13 T 0.413 exp(286/T) cm 3 ?molecule -1 ?s -1 . The experimental data does not extend to a high enough temperature range to indicate either a concave or 82 convex high-temperature deviation of the rate constant from Arhenius behavior. We note that both VariFlex and Polyrate indicate a concave curve for k dis . Generaly, nonlinearity in an Arhenius plot at low temperature is explained by quantum-mechanical tunneling efect and/or the appearance of an additional reaction channel. However in our system, there is no tunneling and no competition betwen two channels. The curvature in the Arhenius plot at high temperatures can be explained by the excitation of vibrational modes. 50,51 As the population of the excited vibrational modes increase with increasing temperature, the reaction probability increases. Therefore, enhancement of the reactivity causes an increase in reaction rate at high temperatures. The individual values of k 1 (k cis ) and k 3 (k trans ) by VariFlex are given in Figure 5 and can be used to compute a cis:trans branching ratio for the initial formation of isomers. At low temperature, the ratio of cis-ONOCl is much greater than trans-ONOCl (1 : 0.65, 200K), but the ratio is reduced at high temperature (1 : 0.83, 500K). ChemRate and Polyrate were used to calculate the isomerization (trans-ONOCl ? cis-ONOCl) rate constant. ChemRate contains a master equation solver so that rate constants for unimolecular reactions in the energy transfer region and chemical activation proceses under steady and non-steady conditions can be determined on the basis of RKM theory. The input from the electronic structure programs was very similar except that CSD(T)/c-pVTZ/CSD(T)/c-pVDZ energies were used in ChemRate while G3B3//B3LYP/6-311+G(d) energies were used in Polyrate. The vibrational frequencies as a function of s for the trans-ONOCl ? cis-ONOCl isomerization reaction are shown in Figure 6 where positive values of s correspond to the 83 5 10 15 20 25 30 1 1.5 2 2.5 3 3.5 4 4.5 5 Polyrate (TST) Polyrate (CVT) Chemrate ln[ k (T)] 1000/T K -1 Figure 7. Comparison of ln(k) versus 1/T for transition state theory (TST) and variational transition state theory (CVT). The ChemRate results are RKM. product side and negative values to the reactant side. The torsion mode is not represented in Figure 6 because it is the imaginary frequency in this range of the reaction coordinate. There is litle variation in reaction coordinate, which indicates that these modes do not have a strong contribution to the reaction dynamics. The Arhenius plots in Figure 7 show that the TST and CVT curves nearly overlap which indicates that the variational efect on the calculation of rate constant is 84 very smal and can be ignored. The rate constant for trans ? cis isomerization (k 4 =1.92x10 13 exp(-4730/T) s -1 ; Polyrate-TST) is found to obey the Arhenius equation at 1 atm in the temperature range of 200-2000K. 3.5 Conclusion The NO + ClO ? NO 2 + Cl reaction has been a chalenge to experiment and theory. This reaction and ones similar to it such as NO + OH, NO + O 2 H, and NO + OF, have been used as a test-bed for computational methods. We have used a series of theoretical methods to elucidate the reaction mechanism. Variational transition state theory was used to compute initial rate constants for the addition reaction to form cis- and trans-ONOCl. The branching ratio favors the cis isomer at lower temperature (1:0.65 at 200K). The N?Cl bond fragmentation is predicted to have a significantly diferent barier in cis-ONOCl (19.75 kcal/mol) compared to trans-ONOCl (31.83 kcal/mol). This diference leads to the interesting prediction that the trans isomer must first isomerize to the cis isomer before a chlorine atom leaves. The rate constant for disappearance of reactants (k dis ) and appearance of products (k obs ) are almost identical and show a pronounced concave curvature indicating non-Arhenius behavior at higher temperature. Over the temperature range 200-400K the activation energy is -0.35 kcal/mol (k dis ). 85 3.6 Reference (1) Wofsy, S. C.; McElroy, M. B. Can. J. Chem. 1974, 52, 1582. (2) Farman, J. C.; Gardiner, B. G.; Shanklin, J. D. Nature 1985, 315, 207. (3) Stolarski, R. S.; Cicerone, R. J. Can. J. Chem. 1974, 52, 1610. (4) Brune, W. H.; Toohey, D. W.; Anderson, J. G.; Chan, K. R. Geophys. Res. Let. 1990, 17, 505. (5) Waters, J. W.; Froidevaux, L.; Read, W. G.; Manney, G. L.; Elson, L. S.; Flower, D. A.; Jarnot, R. F.; Harwood, R. S. Nature 1993, 362, 597. (6) Fickert, S.; Heleis, F.; Adams, J. W.; Moortgat, G. K.; Crowley, J. N. J. Phys. Chem. A 1998, 102, 10689. (7) Carter, R. T.; Halou, A.; Huber, J. R. Chem. Phys. Let. 1999, 310, 166. (8) Clyne, M. A. A.; Watson, R. T. J. Chem. Soc. Faraday Trans. 1974, 70, 2250. (9) Zahniser, M. S.; Kaufman, F. J. Chem. Phys. 1977, 66, 3673. (10) Leu, M. T.; DeMore, W. B. J. Phys. Chem. 1978, 82, 2049. (11) Clyne, M. A. A.; MacRobert, A. J. Int. J. Chem. Kinet. 1980, 12, 79. (12) Ray, G. W.; Watson, R. T. J. Phys. Chem. 1981, 85, 2955. (13) Le, Y.-P.; Stimpfle, R. M.; Pery, R. A.; Mucha, J. A.; Evenson, K. M; Jennings, D. A.; Howard, C. J. Int. J. Chem. Kinet. 1982, 14, 711. (14) Zhao, Y.; Houk, K. N.; Olson, L. P. J. Phys. Chem. A 2004, 108, 5864. (15) Elison, G. B.; Herbert, J. M.; McCoy, A. B.; Stanton, J. F.; Szalay, P. G. J. Phys. Chem. A 2004, 108, 7639. (16) Zhu, R. S.; Lin, M. C. Chem. Phys. Chem. 2004, 5, 1. (17) Bach, R. D.; Dmitrenko, O.; Est?vez, C. M. J. Am. Chem. Soc. 2003, 125, 16204. 86 (18) Dixon, D. A.; Feler, D.; Zhan, C.-G.; Francisco, J. S. J. Phys. Chem. A 2002, 106, 3191. (19) Le, T. J.; Bauschlicher, C. W.; Jayatilaka, D. Theor. Chem. Ac. 1997, 97, 185. (20) Guha, S.; Francisco, J. S. J. Phys. Chem. A 1997, 101, 5347. (21) Francisco, J. S.; Clark, J. J. Phys. Chem. A 1998, 102, 2209. (22) Parthiban, S.; Le, T. J. J. Chem. Phys. 2000, 113, 145. (23) Le, T. J.; Scuseria, G. E. J. Chem. Phys. 1990, 93, 489. (24) Scuseria, G. E. J. Chem. Phys. 1991, 94, 442. (25) Gaussian03, (Revision B.4), Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A. Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Milam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Peterson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Adamo, C.; Jaramilo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cami, R.; Pomeli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Cliford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Chalacombe, M.; Gil, P. M. W.; Johnson, B.; 87 Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian, Inc., Pitsburgh PA, 2003. (26) Karlstr?m, G.; Lindh, R.; Malmqvist, P.-?.; Roos, B. O.; Ryde, U.; Veryazov, V.; Widmark, P.-O.; Cossi, M.; Schimelpfennig, B.; Neogrady, P.; Seijo, L. Computational Material Science 2003, 28, 222. (27) Schaftenaar, G.; Noordik, J. H. J. Comput.-Aided Mol. Design 2000, 14, 123. (28) Woon, D. E.; Dunning, T. H. J. Chem. Phys. 1993, 98, 1358. (29) Kendal, R. A.; Dunning, T. H.; Harison, R. J. J. Chem. Phys. 1994, 100, 7410. (30) Curtis, L. A.; Raghavachari, K.; Redfern, P. C; Rasolov, V.; Pople, J. A. J. Chem. Phys. 1998, 109, 7764. (31) Moore, C. E. Atomic Energy Levels, National Bureau of Standards, Washington, D.C. 1971, Vols. I and I, NSRDS-NBS 35. (32) Coxon, J. A. Can. J. Phys. 1979, 57, 1538. (33) Sayin, H.; McKe, M. L. J. Phys. Chem. A 2004, 108, 7613. (34) The standard G3B3 method includes spin-orbit corrections for atoms but not for diatomic. We have included spin-orbit efects for NO and ClO in the G3B3 energies. (35) Roos, B. O. ?The complete active space self-consistent field method and its applications in electronic structure calculations? In: Lawley, K. P., Ed. Advances in Chemical Physics; Ab Initio Methods in Quantum Chemistry-I; John Wiley & Sons: Chichester, UK, 1987. (36) (a) Roos, B. O.; Anderson, K.; F?lscher, M. P.; Malmqvist, P.-?.; Serano- Andr?s, L.; Pierloot, K.; Merch?n, M. ?Multiconfigurational perturbation theory: 88 Applications in electronic spectroscopy? In: Prigogine, I; Rice, S. A., Ed. Advances in Chemical Physics: New Methods in Computational Quantum Mechanics; John Wiley & Sons: New York, 1995. (b) Andersson, K.; Malmqvist, P.-?.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483. (c) Anderson, K.; Malmqvist, P.-?.; Roos, B. O. J. Phys. Chem. 1992, 96, 1218. (37) (a) Widmark, P. O.; Malmqvist, P.-?.; Roos, B. O. Theor. Chim. Acta 1990, 77, 291. (b) Widmark, P. O.; Malmqvist, P.-?.; Roos, B. O. Theor. Chim. Acta 1991, 79, 419. (38) Corchado, J. C.; Chuang, Y.-Y.; Fast, P. L.; Vila, J.; Hu, W.-P.; Liu, Y.-P.; Lynch, G. C.; Nguyen, K. A.; Jackels, C. F.; Melisas, V. S.; Lynch, I. R.; Coitino, E. L.; Fernandez-Ramos, A.; Pu, J.; Albu, T. V.; Steckler, R.; Garet, B. C.; Isacson, A. D.; Truhlar, D. G. Polyrate, Version 9.3, University of Minnesota, MN, 2004. (39) Klippenstein, S. J.; Wagner A. F.; Dunbar, R. C.; Wardlaw, D. M.; Robertson, S. H. VariFlex, Version 1.00, Argonne National Laboratory, Argonne, IL, 1999. (40) Mokrushin, V.; Bedanov, V.; Tsang, W.; Zachariah, M. R.; Knyazev, V. D. ChemRate, Version 1.19, National Institute of Standards and Technology, Gaithersburg, MD, 2002. (41) Tevault, D. E.; Smardzewski, R. R. J. Chem. Phys. 1977, 67, 3777. (42) Kawashima, Y.; Takeo, H.; Matsumura, C. Chem. Phys. Let. 1979, 63, 119. (43) The NIST Standard Reference Database (htp:/webok.nist.gov/chemistry) was used as the source of al thermochemistry. 89 (44) Zhang, D.; Zhang, R.; Park, J.; North, S. W. J. Am. Chem. Soc. 2002, 124, 9600. (45) Olson, L. P.; Kuwata, K. T.; Bartberger, M. D.; Houk, K. N. J. Am. Chem. Soc. 2002, 124, 9469. (46) Slagle, I. R.; Gutman D.; Davies, J. W.; Piling, M. J. J. Phys. Chem. 1988, 92, 2455. (47) Song, S.; Hanson, R. K.; Bowman, C. T.; Golden, D. M. J. Phys. Chem. A 2002, 106, 9233. (48) Miler, J. A.; Klippenstein, S. J. J. Phys. Chem. A 2000, 104, 2061. (49) Graf, M. M.; Wagner, A. F. J. Chem. Phys. 1990, 92, 2423. (50) Kandel, S. A.; Zare, R. N. J. Chem. Phys. 1998, 109, 9719. (51) Michelsen, H. A. Ac. Chem. Res. 2001, 34, 331. (52) Milen, D. J.; Sinnot, K. M. J. Chem. Soc. 1958, 350. 90 CHAPTER 4 THE DISOCIATION MECHANISM OF A STABLE INTERMEDIATE: PERFLUOROHYROXYLAMINE 4.1 Introduction Perfluorohydroxylamine, F 2 NOF, is an interesting example of an electron-rich molecule that may have a number of competitive rearangement pathways. There are several theoretical studies reported for this molecule; 1,2 semi-empirical, 3 Hartre-Fock methods, 4 and Coupled Cluster theory 1,2 which find that the cis conformation is more stable than trans. While the structure, stability, and thermochemistry of this compound have not been investigated experimentaly, Antoniotti et al. 1 suggested that F 2 NOF is an intermediate in the reaction betwen O( 1 D) and NF 3 which produced F 2 N and FO radicals and Bedzhanyan et al. 6 suggested that F 2 NOF can be an intermediate in the F 2 N + FO reaction. On the other hand, the F 3 NO isomer has been structuraly characterized and posseses a N-O bond with a high degre of double bond character (r N - O =1.159 ?). 5 Antoniotti and Grandineti studied 1 the disociation pathway of F 3 NO at the CSD(T)/aug-c-pVTZ/CSD/c-pVDZ level and found a transition state for the rearangement of trans-F 2 NOF to F 3 NO with a 22.1 kcal/mol enthalpy barier. As 91 discussed below, their transition state corresponds to a higher-lying structure with strong zwiterionic character. The true transition state has much higher radical character. Bedzhanyan et al. 6 studied the reaction betwen F 2 N and FO radicals and found the dominant channel to be eq 1. A reasonable mechanism would have the F 2 N + FO ? FNO + 2F (1) radicals asociate to form F 2 NOF which could then disociate to FNO + 2F through a stepwise cleavage of F atoms. To our knowledge, the experimental vibrational spectrum for F 2 NOF has not been reported. Misochko et al. 7 measured the infrared absorption spectra and EPR spectra of the F 2 NO radical at 20K in an argon matrix, as wel as infrared absorption spectra for FNO, F 2 NO and F 3 NO, but did not identify F 2 NOF. Although postulated as an intermediate in several reaction mechanisms, F 2 NOF has not been identified experimentaly. A reasonable mechanism of F 2 NOF disociation would include eq. 2-6. F 2 NOF ? F 2 N + OF (2) F 2 NOF ? FNO + F 2 (3) F 2 NOF ? F 2 NO + F? (4) F 2 NO ? FNO + F? (5) F? + F? ? F 2 (6) 92 Equation 4, the O-F disociation step, has some similarities with O-X disociations in ONO-X (X=F,Cl,Br,OH,OCl). When X=F, Elison et al. 8 showed that there is direct isomerization betwen cis-ONOF and FNO 2 with an activation energy of 22?3 kcal/mol. When X=Cl, the cis-ONOCl ? ClNO 2 transition state corresponds to fragmentation with a 20.9 kcal/mol barier. 9 Kova?i? et al. 10 recently calculated an isomerization path betwen cis-ONOBr and BrNO 2 with an activation barier of 20.2 kcal/mol. On the other hand, when X=OH, the concerted formation of HNO 3 from HONO is stil not wel established. 1 Zhao et al. 1a showed that there is no direct isomerization betwen HONO and HONO 2 . On the other hand, a recent analysis using master equation simulation 1b found that the best fit with experimental data occurs when the transition state for trans-HONO ? HONO 2 is 5.2 kcal/mol lower in energy than HO + NO 2 . The O-O cleavage occurs from cis-HONO with an activation barier 18-19 kcal/mol to form NO 2 + OH. A similar mechanism of O-O cleavage occurs in the cis-ClONO ? ClONO 2 reaction with an activation energy of 28.4 kcal/mol and 6.7 kcal/mol from studies by Kova?i? et al. 12 and Zhu et al, 13 respectively. Fox et al. 14 reported the synthesis of F 3 NO from FNO plus F in a fluorine-nitric oxide flame. The authors suggested that fluorination of an excited state of F 2 NO (formed in the 2000K flame) might be involved in the mechanism. In this paper, we wil use theoretical methods to calculate the potential energy surface for the disociation of F 2 NOF molecule and calculate the rate constant for formation of products over the temperature range 200-1000K. Al the calculated rate constants reported below are at the high-presure limit. 93 4.2 Computational Method Density functional theory (DFT), widely used as a computational chemistry tool providing reasonable acuracy at modest computational cost, is used in this study because it has been shown to give reasonable structures and vibrational frequencies for halogen compounds. 15-18 We optimized geometries at the B3LYP/6-311+G(d) level, but we checked our results by reoptimizing with B1K 19 and MPWB1K 20 which are hybrid meta DFT methods specificaly designed to yield good results for kinetics. Beside the 6- 311+G(d) 21 basis set, we also used the MG3S 2 basis set which is equivalent to 6- 311+G(2df,2p) for systems without elements heavier than F. We also checked some of the stationary points with multi-configurational SCF to determine the efect of adding additional configurations to the wave function. Lastly, we reoptimized most structures at the CSD/6-31+G(d) level as a further check on consistency of prediction. Since the CSD(T) method with a reasonable basis set yields very good results for diferent chemical systems such as O 3 23 and FOF, 24 we based our kinetics calculations on energies at CSD(T)/c-pVQZ/B3LYP/6-311+G(d) unles indicated otherwise. Al electronic structure calculations have used the Gaussian03 25 and Games 26 program systems. Al imaginary frequencies for transition states were animated by using the graphical program MolDen 27 to make sure that the motion of the transition vector was appropriate for converting reactants to products. The transition states involving bond formation or bond breaking were computed with an unrestricted method (UDFT or UHF) to determine the lower-energy spin broken- symmetry solution at the UDFT or UCSD levels. Single-point calculations were made with a 16-electron 11-orbital complete active space 28 (CASCF(16e,11o)) and the 94 6-311+G(d) basis set with dynamic electron correlation introduced at the MP2 level (MCQDPT2). 29 Optimization of radical fragments and most F 2 NOF stationary points were also caried out at the CASCF(16e,11o) level (Table 2 and Table S5). However, the transition states that were characterized by a loosely asociated F atom (Abst-F 2 -ts-c, Add-F-N-ts, and F 2 -FON-ts) could not be located at the CASCF level probably due to the lack of dynamic electron correlation in the CASCF method. T1 diagnostics were computed at the CSD level (Table 2) for several of the transition states. Values larger than 0.02 are often used as an indicator of significant multireference character. It is noteworthy that F 2 NO-F-ts had a value of 0.02. The intrinsic reaction coordinate 30 (IRC) is constructed starting from the saddle point geometry and going downhil to both the asymptotic reactant and product channels in mas-weighted Cartesian coordinates. Along each IRC, the reaction coordinate, ?s? is defined as the signed distance from the saddle point, with s>0 refering to the product side. Once acurate approximations to the stationary points on the potential energy surface (PES) are available, reaction rate constants can be calculated using variational transition-state-theory (VTST). 31-3 Thre programs were used to compute rate constants. For reactions without a transition structure, Variflex-1.0 34 was used (k 1 , k 5 , k 6 , and k 1 , se below). For the conversion of cis-F 2 NOF to trans-F 2 NOF, Chemrate-1.21 35 was used (k 3 , se below). For other reactions with a transition state structure, Polyrate-9.3 36 was used either with conventional transition state theory (k 4 , k 7 , k 8 , se below) or with variational transition state theory (k 2 , se below). 95 F F N O F 1.30 CASCF/6-31+G(d) .24 B3LYP/ 1.9 1K/MG3S .24 PWB1/3 1.3 CSD/6-3+(d) .0 (T)/c-pVDZ 1.235 S/631+G(d) .64 CD(T)/- f 1.32 .8 1.349 .5 1.391 .2 1.38 .62 108.4 .2 108.9 . 107.9 .4 1.1 2.6 1.463 .59 1.48 .3 1.471 .532 1.64 .52 109.4 2. 10.1 9. 10.6 9. 1.2 .0 F O N F F 1.439 .1 1.378 .5 1.42 .58 1.41 .20 103.9 .5 103.9 . 102. .8 102.6 3.1 1.34 .82 1.35 .0 1.398 .405 1.392 .87 104.1 CASCF/6-31+G(d) 3. B3LYP/ 10.5 1K/MG3S 3. PWB1/3 102. CSD/6-3+(d) .3 (T)/c-pVDZ 102.8 S/631+G(d) 3. CD(T)/- f 1.326 .9 1.351 .48 1.395 . 1.386 .9 F F N O F 1.35 CASCF/6-31+G(d) .4 B3LYP/-() 1.8 1K/MG3S .4 PWB1/3 1.73 CSD/6-3+G(d) .6 (T)/c-pVDZ 1.2 CSD()/6-31+G(d) .64 (T)/-(f) 1.59 Exp a 1.451 . 1.387 .4 1.29 .41 1.36 .4 1.32 Exp a 9. 10.5 .8 10. .6 10.2 .5 10.7 17.9 .4 17.2 .1 17.3 .4 17. .2 17.4 Exp a cis-F 2 NOF trans-F 2 NOF F 3 NO Figure 1. Optimized geometry of cis-, trans-F 2 NOF and F 3 NO isomers. Bond lengths are in ? and angles are in degres. For each isomer, methods are shown with the data. The last row in F 3 NO isomers are experimental values. (a) Frost, D. C.; Hering, F. G; Mitchel, K. A. R; Stenhouse, I. R. J. Am. Chem. Soc. 1971, 93, 1596. 96 Table 1. Harmonic frequencies of cis-F 2 NOF, trans-F 2 NOF and F 3 NO in cm -1 Method cis-F 2 NOF ? 1 (a?) ? 2 (a ? ) ? 3 (a?) ? 4 (a?) ? 5 (a?) ? 6 (a?) ? 7 (a?) ? 8 (a?) ? 9 (a?) B3LYP/6-31+g(d) 1242 914 838 742 584 531 474 207 195 B1K/MG3S 150 1040 94 852 798 595 59 296 215 CSD/6-31+G(d) 1065 975 895 823 73 549 518 27 196 CSD(T)/c-pVDZ 1046 910 850 750 56 547 474 232 194 CSD(T)/6-31+G(d) 1230 951 827 696 578 518 346 189 13 CSD(T)/6-31+G(df) 181 1023 87 748 589 53 385 204 101 trans-F 2 NOF ? 1 (a?) ? 2 (a?) ? 3 (a?) ? 4 (a?) ? 5 (a?) ? 6 (a?) ? 7 (a?) ? 8 (a?) ? 9 (a?) B3LYP/6-31+G(d) 1031 937 86 73 597 474 463 394 53 B1K/MG3S 14 1084 1029 941 68 518 512 431 63 CSD/6-31+G(d) 1058 967 957 876 612 47 471 395 -28 CSD(T)c-pVDZ 107 83 878 764 586 465 460 384 -19 CSD(T)/6-31+G(d) 1028 910 897 797 596 473 467 391 40 CSD(T)/6-31+G(df) 1073 968 951 849 627 489 47 401 43 F 3 NO ? 1 (a 1 ) ? 2 (e) ? 3 (e) ? 4 (a 1 ) ? 5 (a 1 ) ? 6 (e) ? 7 (e) ? 8 (e) ? 9 (e) B3LYP/6-31+G(d) 1787 81 81 758 53 520 520 393 393 B1K/MG3S 1785 974 974 857 609 602 602 438 438 CSD/6-31+G(d) 1735 936 936 78 561 560 560 406 406 CSD(T)/c-pVDZ 1790 89 89 729 534 514 514 397 397 CSD(T)/6-31+G(d) 1764 896 895 742 541 52 52 402 402 CSD(T)/6-31+G(df) 1741 932 932 785 573 56 56 415 415 Exp a 1852 874 741 529 403 (a) Smardzewski R. R.; Fox, W. B. J. Chem. Phys. 1974, 60, 2193. 4.3 Results and Discusions The calculated equilibrium structures of F 3 NO are in good agrement with each other and with experiment (Figure 1). The O-F bond distance of cis-F 2 NOF has the largest sensitivity with respect to method, with CSD(T)/6-311+G(d) giving the longest 97 Table 2. Relative Enthalpies a (kcal/mol) for Various Species Involved in the Disociation of F 2 NOF Molecule /B3LYP/6-31+G(d) b B1K/ MG3S MPWB1K/ G3S CSD/ 6-31+G(d) MCQDPT2/ 6-31+G(d) b,c B3LYP/ 6-31+G(d) MCQDPT2/ 6-31+G(d) CSD(T)/ c-pVTZ d CSD(T)/ c-pVQZ d cis-F 2 NOF 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 trans-F 2 NOF 5.89 5.84 4.51 10.76 9.06 9.36 3.80 4.31 F 3 NO -35.38 -35.36 -28.5 -39.80 -3.98 -3.4 -34.07 -34.63 Abst-F 2 -ts-c e 10.98 8.94 12.05 13.80(0.07) [15.25] g Ad-F-N-ts 27.06 27.69 34.01 21.27 36.06 29.7(0.04) 27.37 F 2 NO-F-ts e 17.15 17.4 18.57 17.31 13.90 15.6 14.1(0.02) 14.10(0.02) F 2 NOF-ts 14.06 14.05 12.37 1.01 16.13 15.35 10.91 1.31 F-F 2 NO-ts e 7.73 f 9.54 1.67 14.25(0.08) [14.51] h F 2 -FON-ts e 18.92 12.27 23.0(0.04) [24.45] g Complex e 8.17 7.56 8.4 16.54(0.04) [9.89] h F 2 NO + F 9.48 10.13 9.12 14.76 10.45 8.23 14.52 15.97 FNO + F 2 -7.79 -6.30 -17.56 -8.23 -10.73 -1.64 -16.46 -15.5 FNO + 2F 16.52 20.48 24.3 17.74 20.54 F 2 N + OF 3.78 34.73 29.41 50.16 35.24 36.03 35.57 38.16 NO + 3F 79.12 71.12 81.98 a Thermodynamic corrections to produce enthalpies at 298K are made from frequencies computed at the given level except where indicated. b Thermodynamic corrections are made at the B3LYP/6-311+G(d) level. c Geometries are optimized at the CASCF(16e,11o)/6-311+G(d) level. d T1 diagnostic at the CSD level is given in parentheses. e Geometries are optimized at the UDFT and UCSD level. Spin-squared values at UB3LYP/6-311+G(d) are 0.93, 0.13, 0.93 0.95 and 0.80 for Abst-F 2 -ts-c, F 2 NO-F-ts, F-F 2 NO-ts, F 2 -FON-ts, and complex, respectively. f Did not fuly met optimization criterion. g The energy at the CSD(T)/c-pVQZ level is estimated by taking the energy diference with F 2 NO + F at the CSD(T)/c-pVTZ level and adjusting to the CSD(T)/c-pVQZ energy of F 2 NO + F (15.97 kcal/mol). h The efect of spin contamination was projected out of the B3LYP/6-311+G(d) energies using the formula E singlet =(2E BS -?E triplet )/(2-), where is the spin-squared value of the singlet broken-symmetry solution (E BS ) and E triplet is the energy of the triplet at the singlet geometry. In addition, the "corrected" energy diference betwen "F-F 2 NO-ts" or "complex" and F 2 NO + F at the B3LYP/6- 311+G(d) level is subtracted from the CSD(T)/c-pVQZ relative energy of F 2 NO + F (15.97 kcal/mol). 98 F F F N O F O N F F 1.643 (.72) [1.85] 1.230 (.5) [1.269] 12. (.3) [10.5] 105.9 (4.8) [105.] 1.39 (.401) [.36] 103.4 (2.6) [10.3] Ad-FN-ts F 2 NO-Fts Abst-F 2 -tsc 2.57 2.694 1.46 1.45 1.63 16.7 102.5 17.8 F N O F F 1.42 (.9) 2.07 (.5) 2.163 (.) 108. (.3) 69. (8.7) 78. (9.4) 1.347 (.2) F O N F F 1.487 (.2) [1.490] 1.4 (.31) [.49] 1.368 (.95) [1.32] 105.9 (6.) [0.5] 103. (2.) [103.4] 109.4 (8.6) [10.4] F 2 NOF-ts F- 2 NO-ts F ON F F 73.4 [80.1] 82.0 [75.] 102.5 [.3] 2.76 [.83] 2.678 [.910] 1.60 [.29] 1.45 [.38] F O N F F 2.184 2.63 2.63 1.26 1.532 1.2 F 2 -NO-ts 1.389 (.75) [1.32] F ON F 1.8 (.61) [.207] 16.9 (.) [15.8] 1.47 (.) [1.347] 102. (.) [105.] N O F 1.543 (.) [1.43] 1.28 (.50) [1.42] 10.4 (9.7) [10.] F F N F 2 NO FNO F 2 N FO 1.35 (.80) [1.3] OF F F 1.409 (.3) [1.507] complex 1.357 (.6) [1.35] 103.5 (2.9) [103.5] F 2 F ON F F 1.70 [.21] 2.086 [.7] 18.6 [3.] 1.423 [.56] 18. [5.7] 103. [5.] Figure 2. Optimized geometric parameters of stationary points at the B3LYP/6-311+G(d) level. Values in parentheses are at the CSD/6-31+G(d) level and values in brackets are at the CASCF/6-311+G(d) level. Bond lengths are in ? and angles are in degres. 99 O-F bond distance and MPWB1K/MG3S the shortest. Likewise, there were large diferences in the N-O bond of cis-F 2 NOF with CSD(T)/6-311+G(d) giving a short N-O bond distance and MPWB1K/MG3S giving a long N-O distance. The B3LYP DFT method made predictions in closest agrement with CSD(T). Table 1 shows the comparison of the calculated frequencies of cis-F 2 NOF, trans- F 2 NOF and F 3 NO. The calculated frequencies are in agrement with each other for cis- F 2 NOF with slightly les agrement for the ONF bend (? 5 ). For F 3 NO, the calculated frequencies are in agrement with each other and very close to experimental results. It is interesting to point out that the CSD(T) method gave one imaginary frequency for trans-F 2 NOF with a smal basis set (c-pVDZ), 37 but al real frequencies with bigger basis sets (6-311+G(d) and 6-311+G(df). A lower-energy spin broken-symmetry solution was obtained at the UDFT and UCSD levels in transition states involving bond formation or bond breaking (Table 2). We checked two of these transition state energies by using MCQDPT2/6- 311+G(d)/UB3LYP/6-311+G(d). Unfortunately we did not get reliable results at this method. The two bonds involving fluorine in the transition state (Abst-F 2 -ts-c) for the reaction cis-F 2 NOF ? FNO + F 2 are very asymmetrical (Figure 2); one bond is almost completely broken (F-O 2.694 ?), while the other (F-N, 1.446 ?) shows no lengthening. Also, the newly forming F-F bond (2.557 ?) is very long. These factors led us to believe that the maximum along the PES might be sensitive to computational method. For these reasons, the rate constant was calculated with VTST using Polyrate. An intrinsic reaction coordinate (IRC) was calculated in mas-weighted coordinates at the B3LYP/6-311+G(d) 100 level. At ten points along the IRC, single-point energies were computed at the UCSD(T)/c-pVTZ, while a generalized normal-mode analysis was performed at the B3LYP/6-311+G(d) level projecting out the reaction coordinate. Despite expectations, the maximum at the UCSD(T)/c-pVTZ level occurred at a path value of s=0, the same as the maximum at the B3LYP/6-311+G(d) level. Because the spin broken-symmetry UCSD(T)/c-pVQZ calculations proved to be too lengthy, we estimated the relative energy of Abst-F 2 -ts-c at the UCSD(T)/c- pVQZ level by taking the Abst-F 2 -ts-c/F 2 NO+F energy diference at the UCSD(T)/c- pVTZ level (0.72 kcal/mol more stable than radicals) and applying it to UCSD(T)/c- pVQZ relative energy of F 2 NO + F. 38 The natural population analysis (NPA) at the UB3LYP/6-311+G(d) level showed that the loosely bound fluorine has nearly a full unpaired electron and has very litle charge which means the transition state is biradical rather than zwiterionic. Unlike the reaction cis-F 2 NOF ? FNO + F 2 , the transition state (F 2 NOF-ts) in the cis-F 2 NOF ? trans-F 2 NOF isomerization did not involve bond forming or breaking, only the rotation of OF about the N-O bond which was about mid-way betwen cis (0 o ) and trans (180 o ). Therefore, we felt that the position of the transition state was probably not sensitive to method and we used normal transition state theory with Chemrate. The calculated enthalpy barier at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) level was 11.31 kcal/mol which is slightly les than the barier for the cis-F 2 NOF ? FNO + F 2 reaction (15.25 kcal/mol, Table 2). The transition state (F 2 NO-F-ts) in trans-F 2 NOF ? F 3 NO was found at the UB3LYP/6-311+G(d), UB1K/MG3S, UMPWB1K/MG3S, and UCSD/6-31+G(d) 101 levels. The O ? F calculated distance in the transition state (Figure 2, F 2 NO-F-ts) is rather short (1.643 ?, UB3LYP/6-311+G(d). Elison et al. 8 also found similar short O ? F distances of 1.726 and 1.693 ? in the trans-ONOF ? FNO 2 transition state at the RCSD(T) and UCSD(T)/c-pVTZ levels, respectively. A similar tight transition state is obtained for O-Cl cleavage in our previous study 9 of trans-ONOCl ? ClNO 2 where a O ? Cl distance of 2.191 and 2.067 ? was calculated in the transition state at the RCSD(T) and UCSD(T)/c-pVDZ levels, respectively. Since litle spin contamination was found at the UB3LYP/6-311+G(d) level (F 2 NO-F-ts, =0.13), restricted CSD (RCSD(T)/c-pVQZ) was used rather than unrestricted CSD. The enthalpy (298K) of the transition state is 1.87 kcal/mol lower than F 2 NO + F which implies the product is a complex rather than fre radicals. We located a complex with a spin-squared value (=0.80) which was 3.45 kcal/mol lower in enthalpy (298K) than F 2 NO-F-ts and 2.89 kcal/mol lower than F 2 NO + F at the B3LYP/6-311+G(d) level. The O-F bond increased from 1.643 ? in the transition state (F 2 NO-F-ts) to 2.086 ? in the complex. From the complex, a second transition state (F- F 2 NO-ts) was reached with an enthalpy of activation of 1.98 kcal/mol and a spin-squared value of 0.93 at the B3LYP/6-311+G(d) level. At the DFT level, the description of the F 2 NOF ? F 3 NO reaction is stepwise via a shalow intermediate which is concerted in the sense that the same fluorine that leaves oxygen adds to the nitrogen as opposed to a fragmentation/recombination mechanism where the fluorine atom added is diferent from the one that is cleaved. Neither restricted nor unrestricted CSD(T) methods do wel at describing biradical character. At the UCSD(T)/c-pVTZ/B3LYP/6-311+G(d) level, the enthalpy 102 of the complex is 2.02 kcal/mol above F 2 NO + F. When we projected out the efect of spin contamination from the DFT energies by an approximate method 39 (se Table 2) and referenced the enthalpy against the CSD(T)/c-VQZ value for F 2 NO + F, the complex was 6.08 kcal/mol more stable than F 2 NO + F radicals. Clearly, the relative enthalpies of the complex and F-F 2 NO-ts are very uncertain. We fel that the enthalpies at the DFT level are too low, but that the enthalpies at UCSD(T) are too high. The electronic nature of the complex is an unsymmetrical two-center thre- electron interaction (2c?3e) betwen the unpaired electron on F and the lone pair on O (F 2 NO?F). The stability of this interaction is known to be exaggerated at the DFT level which is known to have excesive spin and charge delocalization. 40-43 While a stabilization of 6.0 kcal/mol is probably too large, some stabilization is reasonable. Based on our results for the F + FNO complex (se below), we would expect F to bind to F 2 NO with an enthalpy of 1-2 kcal/mol. We decided to model the reaction of F 2 NOF to F 3 NO in two ways. The first model asumes that F 2 NOF pases over F 2 NO-F-ts in competition with fragmentation to F 2 NO + F such that the intermediate and second transition state (F-F 2 NO-ts) are unimportant. In other words, any species that cross the first barier wil form F 3 NO. The second model is the same as the first except that the first transition state (F 2 NO-F-ts) is asumed to lead to an intermediate and second transition state with the same energy. Thus, in the absence of reliable energies for the intermediate and second transition state, we asume the thre structures have the same energy at 0K. In the second model, the intermediate can also fragment to F 2 NO + F which wil reduce the F 3 NO product 103 formation. In our calculations the branching ratio betwen the complex?F 3 NO and complex?F 2 NO+F varied from 86:14 at 298K to 70:30 at 1000K. F F N FO 2 38.16 + F ON F F F 2 -ts 1.3 F FF N O Abst- 2 ts-c 15. trans-F 2 NO 4.31 F ON F F F N O F F Ad-FN-ts 27.3 F ON F 2 NO +F F ON F F F 2 NO-ts 14.0 F- 2 NO-ts [14.5] 15.97 N O F F F F 2 + -15. complex [9.8] -34.6 F 3 NO F F N O F cis- 2 0. F N O F F F 2 -ON-ts 3.91 FNO +2F 0.54 F ON F F F ON F F F O N F F Figure 3. Schematic diagram of the potential energy surface for the disociation of F 2 NOF system computed at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) level. Relative enthalpies are given in kcal/mol at 298K. F 2 N=O + F 3 NOF - NF !* NF "* NF ! F LP NF " NF " F 2 NOF 3 NOF NF "* NF "* NF " F LP NF " NF " Figure 4. Interaction diagram comparing (a) the zwiterionic tansition state (Add-F- N-ts) and (b) the biradical transtion state (F-F 2 NO-ts). 104 In comparing the first model, the rate of product formation is given by k 4 , while in the second model, it is given by k 9 =k 4 k 7 /(k -4 +k 7 ). Since k 7 > k -4 (se Table 7), k 4 ? k 9 . However, the second model also includes fragmentation of the intermediate to F 2 NO + F which acounts for 7% of products at 298K in our calculations. Table 3. Spin Density, Natural Population Analysis (NPA) and Geometry are calculated at the (U)B3LYP/6-311+G(d) Spin Density NPA charge Geometry F O N F O N N-O O-F N-F trans-F 2 NOF 0.00 0.00 0.00 0.00 -0.11 0.01 0.56 1.38 1.43 2.21 F 2 NO-F-ts 0.13 0.32 -0.12 -0.13 -0.18 -0.03 0.61 1.23 1.64 2.31 complex 0.81 0.82 -0.23 -0.41 -0.16 -0.14 0.70 1.17 2.08 2.69 F-F 2 NO-ts 0.93 0.94 -0.29 -0.41 -0.05 -0.17 0.68 1.16 2.68 2.77 F 3 NO 0.00 0.00 0.00 0.00 -0.22 -0.22 0.87 1.15 2.22 1.44 Add-F-N-ts 0.00 0.00 0.00 0.00 -0.61 -0.01 0.84 1.14 2.07 2.16 Antoniotti et al. 1 found a transition state for the reaction trans-F 2 NOF ? F 3 NO with a barier of 22.2 kcal/mol at the CSD(T)/aug-c-pVTZ/CSD/c-pVDZ level. In the present study, we find a zwiterionic transition state with a barier of 23.06 kcal/mol (27.37-4.31, Table 2) which is very similar to the one found by Antoniotti et al. However, we also find a lower barier of 9.79 kcal/mol through a biradical transition state (F 2 NO-F-ts). The relevant interactions betwen F and F 2 NO in the two transition state are given in Figure 4. In the zwiterionic transition state (Add-F-N-ts, Figure 4a), the migrating F atoms has a large negative charge (0.61 e - ). The nitrogen center is planar in the F 2 NO fragment as expected for F 2 N=O + . The fluoride sits above the N=O double 105 bond and adds to the nitrogen side to form F 3 NO. In the biradical transition state (F- F 2 NO-ts, Figure 4b), the migrating F atom has litle charge but large unpaired spin density (Table 3). The nitrogen center is pyramidal, as expected for a F 2 NO? radical. The reason for the large activation energy diference can be explained by the electronic excitation (Figure 4). The ?-spin electron is excited from an a? orbital to an a? orbital which leaves an unpaired electron on nitrogen in an orbital which is suitable for bond formation with a fluorine atom. The energy needed for electronic excitation is the reason for the greater energy of Add-F-N-ts compared to F 2 NO-F-ts. The reaction of trans-F 2 NOF ? F 3 NO can be considered a 1,2-fluorine shift which can be compared with the 1,2-hydrogen shift of NH 2 OH. In their study of trans- NH 2 OH ? H 3 NO, Bach et al. 4 found a transition state with a 55.9 kcal/mol barier at the MP4/6-31G(d)/MP2/6-31G(d) level, a barier significantly lower than the H 2 NO-H bond energy of 76.5 kcal/mol (Table 4). In the 1,2-fluorine shift, the barier and F-O bond energies are nearly the same (Figure 3). Table 4. Enthalpies and fre energies of Fluorine loss reaction at the CSD(T)/c- pVQZ/B3LYP/6-311+G(d) level reaction ?H(298K) ?G(298K) Dixon et. al. ?H(0K) a cis-F 2 NOF ? F 2 NO + F 15.97 6.54 H 2 NOH ? H 2 NO+H 76.5 cis-F 2 NOF ? FNO + 2F 20.54 2.56 cis-F 2 NOF ? NO + 3F 81.98 56.30 F 2 NO ? FNO + F 4.57 -3.98 H 2 NO ? HNO+H 61.1 FNO ? NO + F 61.44 53.75 HNO ? NO+H 47.0 (a) Dixon, D. A.; Francisco, J. S.; Alexeev, Y. J. Phys. Chem. A 2006, 110, 185. 106 In their study of the ClONO ? ClONO 2 reaction, Kova?i? et al. 12 calculated an activation energy of 28 kcal/mol at the B3LYP/6-311G(d) level. The same reaction was found to have a much lower activation energy of 6.7 kcal/mol in a study in Zhu et al. 13 at the CSD(T)/6-311+(3df)/B3LYP/6-311+G(3df) level. The 6.7 kcal/mol barier (0K) for the ClONO ? ClONO 2 reaction is much more consistent with our 9.79 kcal/mol barier (298K) for the trans-F 2 NOF ? F 3 NO reaction since the propensity for migrating a OCl radical should be similar to migrating a F radical. The calculated bond enthalpy 45 at 298K for ClO-NO 2 (28.0 kcal/mol) and BrO- NO 2 (28.9 kcal/mol) at the CSD(T) are in good agrement with experimental results (26.8 kcal/mol and 28.2 kcal/mol, respectively). We used the same method to find the FO-NF 2 bond enthalpy of 38.16 kcal/mol at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) level. Table 5. Enthalpies of the various types of F 2 NO species optimized at the CSD/c- pVDZ level ?H(0K) ?H(298K) CCSD/ cc-pVDZ CCSD(T)/ cc-pVTZ CCSD(T)/ cc-pVQZ CCSD/ cc-pVDZ CCSD(T)/ cc-pVTZ CCSD(T)/ cc-pVQZ F 2 NO a 0.0 0.0(0.0) 0.0(0.0) 0.0 0.0(0.0) 0.0(0.0) t-F-FNO-c s a -6.95 1.24(1.35) 2.79(2.86) -5.73 2.46(2.57) 4.01(4.08) c-F-FNO-c s -6.89 1.16 2.75 -5.71 2.34 3.93 F-FNO-ts 3.38 5.67 5.6 3.4 5.75 5.74 FNO+F -5.8 2.05(1.95) 3.57(3.4) -4.75 3.18(3.01) 4.70(4.51) a) The data in the parentheses are at CSD(T)/c-pVDZ optimized geometries. 107 4.4 Stability of F 2 NO The lowest energy fragmentation for cis-F 2 NOF produces F 2 NO + F radicals. The lifetime of a thermalized cis-F 2 NOF molecule calculated from k 1 is 2.5x10 -3 s at 298K. However, the N-F bond enthalpy (298K) is very smal (4.57 and 3.57 kcal/mol at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) and CSD(T)/c-pVQZ/CSD/c-pVDZ levels, respectively, Table 4 and 5). In fact, the fre energy change at 298K is spontaneous (-3.98 kcal/mol) for breaking the N-F bond in F 2 NO (Table 4). It is interesting that while the N-F bond in F 2 NO is very weak, the N-F in FNO is very strong. Exactly the opposite behavior is calculated for the N-H bonds in H 2 NO and HNO where the first is 61.1 kcal/mol and the second one is 47.0 kcal/mol at 0K (Table 4). Fluorine is known to be a very good ? donating substitutent. In FNO, fluorine can donate electron density into the ?* orbital of NO while hydrogen cannot donate in HNO. In contrast to F 2 NOF and in spite of its lack of thermal stability, the difluoronitroxide radical (F 2 NO) has been produced in solid argon matrices by addition of F atoms to NO by Misochko et al. 7 They found that F 2 NO exists in equilibrium (F-FNO ? ? ? F 2 NO) with a van der Wals complex (F-FNO). Since F 2 NO and the complex were formed at very low temperature (20K), we compared our results at 0K with experimental results at Table 5. The reaction mechanism of F 2 NO ? FNO + F are calculated at the CSD(T)/c-pVQZ/CSD/c-pVDZ in Figure 5. The complex (F-FNO) and transition state (F-FNO-ts) are located at the CSD/c-pVDZ level. The optimized geometries of the complex and transition state of B3LYP/6-311+G(d) level are compared with CSD/c-pVDZ in Figure 6. The CSD(T)/c-pVDZ method was used to optimized 108 geometries for the complex (t-F-FNO-c s ) and F 2 NO and found to give very similar geometries with the CSD/c-pVDZ method. While B3LYP gives reasonable structures for halogen compounds, it is interesting that B3LYP failed for the complex. The reason for this can be understood by excesive spin delocalization in DFT that is a particular problem in asymmetric 2c-3e bonding 40-43 (Table 6). F 2 NO F-NO-ts t-FNO-c s FNO + F 0. (.) 5.6 2.79 (.86) 3.57 (.4) F ON F F O N F N O F F F O N Figure 5. Schematic diagram of the potential energy surface ?H(0K) for the disociation of F 2 NO at the CSD(T)/c-pVQZ/CSD/c-pVDZ level. Values in parentheses are at the CSD(T)/c-pVQZ/CSD(T)/c-pVDZ level. We optimized two F-FNO complexes at the CSD/c-pVDZ level (c-F-FNO-c s and t-F-FNO-c s , Table 5) and one at the CSD(T)/c-pVDZ level (t-F-FNO-c s ). The cis complex is very slightly more stable (by 0.04 kcal/mol at 0K) but the trans complex should be formed first by the principle of least motion. Both cis and trans complexes are very diferent from the complex obtained at the DFT level (Figure 6). In the radical 109 O F N F 2.089 3.069 1.2 1.643 87.4 109.7 5.2 F O N F 1.52 1.73 1.43 18.7 103.8 t-FNO-c s F-NO-c 1 DT F-NO-ts F-NO-ts1 DT c-FNO-c s ON F F 1.4 1.495109. 10.5 4.8 3.04 2.510 F ON F 1.83 (7) 1.426 () 16.8 (71) 102. (8) F 2 NO F O N F 2.643 2.810 1.54 1.26 132. 79.6 7. F F O N 1.43 (2) 1.495 (31) 2.539 (41) 3.102 (75) 09. (12) 97.1 (68) 54.3 (6) Figure 6. Optimized geometric parameters of stationary points at the CSD/c-pVDZ level. The parameters in parentheses are optimized at the CSD(T)/c- pVDZ level. Table 6. Spin Densities and Mulliken Charges for c-F-FNO-c s and F-FNO-c 1 CSD/c-pVDZ/ CSD/c-pVDZ B3LYP/6-311+G(d)/ B3LYP/6-311+G(d) c-F-FNO-c s F-FNO-c 1 Spin density Mulliken Charges Spin density Mulliken Charges O 0.00 0.00 -0.02 0.24 N 0.00 0.29 -0.00 0.06 F 0.01 -0.28 0.17 -0.18 F 0.99 -0.01 0.85 -0.12 110 complex betwen FNO and F at the DFT level, the interaction is betwen the unpaired electron on one fluorine and the lone pair of another fluorine. The oxygen lone pair electrons in FNO have been stabilized (relative to F 2 NO) which leads to a stronger interaction with the lone pairs on F of FNO. We calculated ?H o =-2.75 kcal/mol enthalpy and ?S o =-13.76 cal/mol?K entropy for the reaction F-FNO ??? ? ? F 2 NO at 0K (Table 5). Experimental results show that the changes in enthalpy and entropy at 20K for the equilibrium reaction (F-FNO ??? ? ? F 2 NO) are 0.29 kcal/mol and 14.82 cal/mol?K, respectively. However, it is not possible to make a direct comparison betwen experiment and theory because the calculations model the gas phase while experiment takes place in an Ar matrix. A greater stabilization energy for the complex could be rationalized by greater dipole-induced dipole interactions betwen Ar molecules and the complex, since the complex has a higher dipole moment than F 2 NO (1.548 and 0.413 D, respectively) at the CSD/c-pVDZ level and dipole- induced dipole interactions are directly proportional to the dipole moment. The lifetime for F 2 NO is only 2.69x10 -10 s -1 at 298K from k 8 . 4.5 Rate Constant Calculations Since there is no reverse barier for cis-F 2 NOF ? F 2 NO + F (k 1 ) and trans- F 2 NOF ? F 2 NO + F (k 5 ), we used Variflex to calculate the rate constants (se eqs. 7-10). The reactive flux was evaluated by the phase-space-integral based VTST (PSI-VTST) method, as implemented in VariFlex 34 as the O-F distances increased from 1.5 to 4.0 ? 111 with a step size 0.1 ? for cis-F 2 NOF ? F 2 NO + F and trans-F 2 NOF ? F 2 NO + F reactions. cis-F 2 NOF FNO +F 2 (concerted) (8) 2 7cis-F 2 NOF k 1 2 trans-F 2 NOFF 2 NO +F F 3 NO (9) trans-F 2 NOF F 3 NO (10) k 5 k -5 [k 6 ] complex k 4 k -4 k 7 FNO +F k 8 F 2 NO +F k 1 k 9 k 10 [k 12 ] k 3 k -3 cis-F 2 NOF Chemrate was used to calculate the isomerization (cis-F 2 NOF "? trans-F 2 NOF) rate constants (k 3 , k -3 ). Chemrate includes a master equation solver so that rate constants for unimolecular reactions can be determined on the basis of RKM theory in the energy transfer region and chemical activation proceses under steady and non-steady conditions. CSD(T)/c-pVQZ/B3LYP/6-311+G(d) energies are used with B3LYP/6- 311+G(d) optimized geometries and thermal corrections in Chemrate. Figure 7 compares the calculated rate constants (high-presure limit) for production of F 2 NO + F (k 1 ) and FNO + F 2 (k 2 ). Since TST rate constants and VTST rate constants are similar for cis-F 2 NOF ? FNO + F 2 (concerted) as the temperature 112 5 10 15 20 25 1 2 3 4 5 trans-F 2 NOF -> F 3 NO (k 9 ) concerted trans-F 2 NOF -> F 3 NO (k 1 2 ) non-concerted trans-F 2 NOF -> cis-F 2 NOF (k - 3 ) trans-F 2 NOF -> F 2 NO + F (k 5 ) l n [ k ( T ) ] 1000/T Figure 7. Calculated rate constants involving cis-F 2 NOF. The computational rate data are for the unimolecular rate constant at the high-presure limit. 113 decreases, variational efect can be ignored for this reaction. The rate constant for cis- F 2 NOF ? F 2 NO + F and cis-F 2 NOF ? FNO + F 2 are calculated as k 1 =6.37x10 13 ?exp(- 7855/T) and k 2 =8.14x10 13 ?exp(-7860/T), respectively. Table 7. Rate constant with temperature dependence at high-presure limit for formation of F 3 NO and disociation of complex to radicals Temp(K) k 4 k -4 k 7 k 1 200 4.45E+01 1.04E+12 1.09E+14 2.03E+12 220 4.68E+02 1.01E+12 1.01E+14 2.71E+12 250 7.95E+03 9.78E+11 9.22E+13 3.87E+12 298 2.28E+05 9.29E+11 8.40E+13 5.95E+12 333 1.46E+06 8.99E+11 8.02E+13 7.60E+12 380 1.01E+07 8.66E+11 7.69E+13 9.85E+12 400 2.03E+07 8.54E+11 7.58E+13 1.08E+13 500 2.83E+08 8.05E+11 7.23E+13 1.56E+13 600 1.65E+09 7.74E+11 7.05E+13 2.00E+13 700 5.83E+09 7.52E+11 6.94E+13 2.40E+13 800 1.51E+10 7.37E+11 6.87E+13 2.76E+13 900 3.15E+10 7.26E+11 6.83E+13 3.07E+13 1000 5.69E+10 7.18E+11 6.80E+13 3.35E+13 The rate constant for formation of F 3 NO is derived with a steady state approximation [k 10 =k 3 k 4 k 7 /(k -3 (k -4 +k 7 )] via trans-F 2 NOF in eq 10. Since the F 2 NO-F-ts transition state is calculated with an acurate method (RCSD(T)/c- pVQZ/UB3LYP/6-311+G(d)), we used this transition state to calculate a rate constant of 1.42x10 12 ?exp(-7420/T) for k 10 . 114 5 10 15 20 25 1 2 3 4 5 trans-F 2 NOF -> F 3 NO (k 9 ) concerted trans-F 2 NOF -> F 3 NO (k 1 2 ) non-concerted trans-F 2 NOF -> cis-F 2 NOF (k - 3 ) trans-F 2 NOF -> F 2 NO + F (k 5 ) l n [ k ( T ) ] 1000/T Figure 8. Calculated rate constants involving trans-F 2 NOF. 115 We also checked our rate constant result corresponding to eq 10 with a steady state approximation for the complex [k 9 =k 4 k 7 /(k -4 +k 7 )]. Since the complex and F 2 -NOF- ts could not be calculated acurately due to spin contamination, we asume that complex and F-F 2 NOF-ts have the same energy as F 2 NO-F-ts in the rate constant calculations. Rate constants k -4 and k 7 are calculated by Polyrate and k 8 is calculated by Variflex. Since k -4 is smal and k 7 is large (k 7 >k -4 ), the overal rate becomes k 9 ?k 4 . We also include the possibility of disociation of the complex (k 8 ) to radicals (F 2 NO + F) in the rate constant for F 3 NO formation. Table 7 shows that k 7 is 14 times faster than the rate constant of complex disociation to radicals (k 1 ) at room temperature which wil reduce the formation of F 3 NO seven percent at room temperature. At al temperatures the ratio of F 2 NO + F (k 1 ) formation to FNO + F 2 (k 2 ) is very similar but much faster than F 3 NO formation. Since F 2 NO has a very short lifetime, it can quickly disociate to FNO + F. Depending on the rate of 2F radical-radical recombination, the dominant products from F 2 NOF decomposition are expected to be FNO, F 2 , and F. It is interesting to note that Fox et al. 14 suggested that the observed formation of F 3 NO from the reaction of F 2 plus NO in a 2000K flame probably involved an excited-state intermediate. While the addition of F to F 2 NO does not play a role in F 3 NO formation, 46 we compared this path (non-concerted path, k 12 ) with the concerted path (k 9 ) of trans-F 2 NOF ? F 3 NO that can be formed with activation enthalpy 14.10 kcal/mol. Our calculated results show that the concerted path (k 9 ) is faster than the non-concerted path (k 12 ) for trans-F 2 NOF ? F 3 NO at low temperatures (Figure 8). This result may have implications in reactions where the radicals generated have longer lifetimes, such as in the FONO ? 116 FNO 2 rearangement. In that reaction, a radical/radical complex may lead to a concerted/nonconcerted branching ratio. 4.6 Conclusion Perfluorohydroxylamine (F 2 NOF) is a chalenging molecule for theory and its short lifetime suggests that it wil be a chalenge for experiment as wel. The O-F bond enthalpy (298K) is calculated at the CSD(T)/c-pVQZ/B3LYP/6-311+G(d) level to be 15.97 kcal/mol. In competition with O-F bond fragmentation, cis-F 2 NOF can eliminate F 2 or isomerize to trans-F 2 NOF. In turn, trans-F 2 NOF can cleave the O-F bond or rearange to F 3 NO via an intermediate F+F 2 NO complex. Rate constants have been calculated for the diferent pathways in order to determine the products formed. At room temperature, only 3% of products (k 10 /(k 1 +k 2 +k 10 ) is expected to be F 3 NO even though it is the global minimum. 117 4.7 Reference (1) Antoniotti, P.; Grandineti, F. Chem. Phys. Let. 2002, 366, 676. (2) Erben, M. F.; Diez, R. P.; V?dova, C. O. D. Chem. Phys. 2005, 308, 193. (3) Dewar, M. J. S.; Rzepa, H. S. J. Am. Chem. Soc. 1978, 100, 58. (4) (a) Olsen, J. F.; Howel, J. M. J. Fluorine Chem. 1978, 12, 123. (b) Olsen, J. F.; O?Connor, D.; Howel, J. M. J. Fluorine Chem. 1978, 12, 179. (5) Plato, V.; Hartford, W. D.; Hedberg, K. J. Chem. Phys. 1970, 53, 3488. (6) Bedzhanyan, Yu. R.; Gershenzon, Yu. M.; Rozenshtein, V. B. Kinetika i Kataliz, 1990, 31, 1474. (7) (a) Misochko, E. Ya.; Akimov, A. V.; Goldschleger, I. U. J. Am. Chem. Soc. 1998, 120, 11520. (b) Misochko, Yu. R.; Akimov, A. V.; Goldschleger, I. U.; Boldyrev, A. I.; Wight, C. A. J. Am. Chem. Soc. 1999, 121, 405. (8) Elison, G. B.; Herbert, J. M.; McCoy, A. B; Stanton, J. F.; Szalay, P. G. J. Phys. Chem. A 2004, 108, 7639. (9) Sayin, H.; McKe, M. L. J. Phys. Chem. A 2005, 109, 4736. (10) Kova?i?, S.; Lesar, A.; Hodo??ek, M. Chem. Phys. Let. 2005, 413, 36. (11) (a) Zhao, Y.; Houk, K. N.; Olson, L. P. J. Phys. Chem. A 2004, 108, 5864. (b) Zhang, J.; Donahue, N. M. J. Phys. Chem. A 2006, ASAP. (c) Srinivasan, N. K.; Su, M.-C.; Sutherland, J. W.; Michael, J. V.; Ruscic, B. J. Phys. Chem. A 2006, ASAP. (d) Hippler, H.; Krasteva, N.; Nasterlack, S.; Striebel, F. J. Phys. Chem. A 2006, ASAP. (12) Kova?i?, S.; Lesar, A.; Hodo??ek, M. J. Chem. Inf. Model. 2005, 45, 58. (13) Zhu, R. S.; Lin, M. C. Chem. Phys. Chem. 2005, 6, 1514. 118 (14) Fox, W. B.; Sukornick, B.; Mackenzie, J. S.; Sturtevant, R. L.; Maxwel, A. F.; Holmes, J. R. J. Am. Chem. Soc. 1970, 92, 5240. (15) Le, T. J.; Bauschlicher, C. W., Jr.; Jayatilaka, D. Theor. Chem. Ac. 1997, 97, 185. (16) Guha, S.; Francisco, J. S. J. Phys. Chem. A 1997, 101, 5347. (17) Francisco, J. S.; Clark, J. J. Phys. Chem. A 1998, 102, 2209. (18) Parthiban, S.; Le, T. J. J. Chem. Phys. 2000, 113, 145. (19) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 2715. (20) Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, 6908. (21) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: NewYork, 1987. (22) Lynch, B. J.; Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2003, 107, 1384. (23) Le, T. J.; Scuseria, G. E. J. Chem. Phys. 1990, 93, 489. (24) Scuseria, G. E. J. Chem. Phys. 1991, 94, 442. (25) Gaussian03, (Revision B.4), Frisch, M. J. et al. Gaussian, Inc., Pitsburgh PA, 2003. (26) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347. (27) Schaftenaar, G.; Noordik, J. H. J. Comput.-Aided Mol. Design 2000, 14, 123. (28) Roos, B. O. ?The complete active space self-consistent field method and its applications in electronic structure calculations? In: Lawley, K. P., Ed. Advances 119 in Chemical Physics; Ab Initio Methods in Quantum Chemistry-I; John Wiley & Sons: Chichester, UK, 1987. (29) Nakano, H. J. Chem. Phys. 1993, 99, 7983. (30) Gonzalez, C.; Schlegel, H. B. J. Phys. Chem. 1989, 93, 2154. (31) Garet, B. C.; Truhlar, D. G. J. Chem. Phys. 1984, 81, 309. (32) Truhlar, D. G.; Garet, B. C.; Klippenstein, S. J. J. Phys. Chem. 1996, 100, 12771. (33) Chuang, Y.-Y.; Corchado, J. C.; Truhlar, D. G. J. Phys. Chem. A 1999, 103, 1140. (34) Klippenstein, S. J.; Wagner A. F.; Dunbar, R. C.; Wardlaw, D. M.; Robertson, S. H. VariFlex, Version 1.00, Argonne National Laboratory, Argonne, IL, 1999. (35) Mokrushin, V.; Bedanov, V.; Tsang, W.; Zachariah, M. R.; Knyazev, V. D. ChemRate, Version 1.21, National Institute of Standards and Technology, Gaithersburg, MD, 2002. (36) Corchado, J. C.; Chuang, Y.-Y.; Fast, P. L.; Vila, J.; Hu, W.-P.; Liu, Y.-P.; Lynch, G. C.; Nguyen, K. A.; Jackels, C. F.; Melisas, V. S.; Lynch, I. R.; Coitino, E. L.; Fernandez-Ramos, A.; Pu, J.; Albu, T. V.; Steckler, R.; Garet, B. C.; Isacson, A. D.; Truhlar, D. G. Polyrate, Version 9.3, University of Minnesota, MN, 2004. (37) Dunning-type basis sets (a) c-pVDZ: Woon, D. E.; Dunning, T. H. J. Chem. Phys. 1993, 98, 1358. (b) c-pVTZ: Kendal, R. A.; Dunning, T. H.; Harison, R. J. J. Chem. Phys. 1992, 96, 6796. (c) c-pVQZ: Dunning T, H. J. Chem. Phys. 1989, 90, 1007. 120 (38) When calculating the rate constant using VTST, the energy of the points along the IRC at the UCSD(T)/c-pVTZ level were shifted upward by 0.72 kcal/mol to be consistent with the correction for the Abst-F 2 -ts-c transition state. (39) (a) Noodleman, L.; Case, D. A. Adv. Inorg. Chem. 1992, 38, 423. (b) Se: Baly, T.; Borden, W. T. in Reviews Comp. Chem. K. B. Lipkowitz, D. B. Boyd, Eds., Wiley: New York, 1999; Vol. 13, p. 1. (40) Fourr?, I.; Silvi, B.; Sevin, A.; Chevreau, H. J. Phys. Chem. A 2002, 106, 2561. (41) Bra?da, B.; Thogersen, L.; Wu, W.; Hiberty, P. C. J. Am. Chem. Soc. 2002, 124, 11781. (42) Bra?da, B.; Hiberty, P. C. J. Phys. Chem. A 2000, 104, 4618. (43) Bra?da, B.; Hiberty, P. C.; Savin, A. J. Phys. Chem. A 1998, 102, 7872. (44) Bach, R. D.; Owensby, A. L.; Gonzalez, C.; Schlegel, H. B.; McDoual, J. J. W. J. Am. Chem. Soc. 1991, 113, 6001. (45) Zou, P.; Derecskei?Kovacs, A.; North, S. W. J. Phys. Chem. A 2003, 107, 888. (46) If we compare k 8 /[F] and k 6 at 298K where [F] is asumed to be 0.01 atm, then the unimolecular rate (k 8 [F 2 NO]) is stil 2300 times faster than the bilmolecular rate (k 6 [F 2 NO][F]).