Ind. Eng. Chem. Res. 2006, 45, 2981-2985
2981
First-Principle Kinetic Modeling of the 1-Chloroethyl Unimolecular Decomposition Reaction Xiaobo Zheng and Paul Blowers* Department of Chemical and EnVironmental Engineering, The UniVersity of Arizona, P.O. Box 210011, Tucson, Arizona 85721-0011
In this work, Gaussian-3 (G3) and complete basis set (CBS) composite energy methods combined with canonical transition state theory (CTST) and Rice-Ramsperger-Kassel-Marcus (RRKM) theory were applied to the kinetic modeling of the 1-chloroethyl radical decomposition reaction: *CHClCH3 f CHClCH2 + *H. Experimental thermodynamic and kinetic data were employed to evaluate the accuracy of these two composite energy methods. For this reaction, both composite energy methods proved to have good agreement with the experimental data, which indicates they are reliable methods for studying reactions involving other chlorinated hydrocarbons (CHCs). A kinetic model of this reaction with pressure and temperature effects was proposed. For P < P0, k (s-1) ) (7.32 × 1010)P0.69 e(-18 727.10/T); for P g P0, k (s-1) ) (3.74 × 1013) e(-20 648.10/T), where P is pressure in units of kPa, T is temperature in units of Kelvin, and P0 ) (8.42 × 103) e(-2 782.61/T). Without performing additional costly calculations, these equations can be easily implemented in different industrial applications with various reaction conditions. 1. Introduction Chlorinated hydrocarbons (CHCs) are important species in the treatment of hazardous compounds with combustion processes.1 They are also crucial for the incineration of natural gas.2 Therefore, the understanding of CHC reaction energetics and kinetic information has broad application in the combustion and petrochemical industries. In general, radicals are highly reactive species with very short lifetimes, and the product radicals are difficult to isolate before reacting further, which makes experimental study of their reaction kinetics very difficult. The experimental kinetic information for hydrocarbon radical cracking reactions is only available for several simple species.3-5 Even these experimental data are limited to moderate temperatures. The focus of this work will be on the 1-chloroethyl radical decomposition reaction,
*CHClCH3 f CHClCH2 + *H where * denotes an unpaired electron. This is a reaction where some limited experimental information is available.5,6 Therefore, this reaction can be applied as a benchmark to evaluate the accuracy of various theoretical methodologies used in this work. With the accuracy of methods proven by this reaction, the methods can be implemented for other chlorinated hydrocarbon species where experimental data are currently unavailable. Knyazev et al. have studied the kinetics of the decomposition reaction using the master equation approach.5 The master equation approach is one method to treat collisional energy transfer to obtain unimolecular reaction rates. 〈∆E〉down, the average energy transferred in deactivations, was derived with a large uncertainty of (30%, and it requires the high-pressurelimit experimental reaction rates. 〈∆E〉down greatly depends on experimental conditions and constrains the ability of extending their results to other conditions outside the scope of their work. In this work, a generalized kinetic model is proposed which * Corresponding author. Tel.: 1-520-626-5319; Fax: 1-520-6216048; E-mail address:
[email protected] (Paul Blowers).
can be applied to different reaction conditions while reducing the uncertainties in the predictions. 2. Computational Method 2.1. Composite Energy Method. First-principle ab initio quantum chemistry methods have long been applied as a major tool for investigating the structure, stability, reaction kinetics, and mechanisms of different molecular systems.7-16 Ab initio calculations are based on the Schro¨dinger equation and have the advantage of depending only on the fundamental laws of physics and universal constants. Therefore, no empirical constants are required in the calculations. Post self-consistent field (SCF) correlation Møller-Plesset (MP) perturbation theory methods have been proven to predict accurate structures of both stable structures and transition states.17,18 In this work, the second-order Møller-Plesset perturbation theory method,19 MP2, is used to perform the geometry optimizations. Electronic structure energy calculations traditionally consist of a single computation. However, to reach high accuracy compared to experimental molecular structures or energetics, the calculation generally requires a very large basis set and a high-level method, which takes a significantly long time to complete. Composite energy methods, also referred to as compound models, were proposed to reach a high level of energetic accuracy at a reduced computational cost. They are defined as a series of single-point calculation steps whose results are then combined to obtain the final electronic energy value. For example, the Gaussian-3 (G3) method developed by Pople and co-workers has shown great promise for predicting heats of reaction, ionization potentials, and other phenomena at a relatively low computational cost compared to its G2 ancestor.20-24 However, the single-point calculation using the G3 large basis set is still very expensive for larger species of interest. More recently, another series of compound models named the complete basis set (CBS) methods has been developed.25-33 These methods eliminate some of the empirical correlations that are included in the Gaussian-n series of methods while still giving very accurate predictions of heats of formation and enthalpies of reaction. Zheng and Blowers34 proposed the CBS-
10.1021/ie051403g CCC: $33.50 © 2006 American Chemical Society Published on Web 04/04/2006
2982
Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006
RAD(MP2) composite energy method as a modification to the computationally very expensive CBS-RAD method, which works especially well for free radicals with high spin contamination effects (RAD refers to radicals). The CBS-RAD(MP2) model replaces the time-consuming QCISD(fc)/6-31g* geometry optimization and frequency calculation method in the CBS-RAD method with the MP2(full)/6-31g* method and basis set, which provides similar accuracy at a reduced computational cost, only 32% of the computational cost of the G3 method.34 Hereafter, the CBS-RAD(MP2) method will be referred to as CBS. In this work, the G3 and CBS compound models were used to study the reaction energetics and kinetics of the 1-chloroethyl radical decomposition reaction. All of the ab initio calculations were performed with the Gaussian 98 software package.35 Geometries were fully optimized at the MP2(full)/6-31g* level. All products and reactants were verified with frequency calculations to be stable structures, and all transition states were determined to be first-order saddle points with only one negative eigenvalue. Additionally, intrinsic-reaction-coordinate (IRC)36 calculations showed that the transition state structure linked the correct products with reactants. Zero point vibrational energies (ZPVE) were obtained from harmonic vibrational frequencies calculated at the MP2(full)/6-31g* level with a scaling factor of 0.9661, and frequencies were scaled with a factor of 0.9427 at the MP2(full)/6-31g* level.37 These frequencies were used in the partition functions for the prediction of reaction rate constants. Also, thermal corrections were included in addition to the ZPVE at all temperatures where reaction rate constants were calculated. 2.2. Rice-Ramsperger-Kassel-Marcus (RRKM) Theory. Rice-Ramsperger-Kassel-Marcus (RRKM) theory is the most commonly used method for predicting reaction rate constants for unimolecular reactions of polyatomic molecules.38 The reaction rate constant for a unimolecular reaction can be modeled as
LQ+ 1 exp(-E0/kT) kuni ) hQ1Q2
W(E+) exp(-E+/kT) dE+ E+)0 1 + ka(E*)/βcZLJ[M] (1)
∫
∞
ka(E*) )
+ LQ+ 1 W(E )
hQ1F(E*)
(2)
where L is the statistical factor; E* is the total vibrational and rotational energy; E + is the total energy of a given transition state; E0 is the activation energy; Q+ 1 is the partition function for the rotation of transition state A+; Q1 is the partition function for the rotation of reactant A; Q2 is the partition function for nonrotational modes of reactant A; βc is the collision efficiency; ZLJ is the Lennard-Jones collision frequency; k is the Boltzmann’s constant; h is the Plank’s constant; [M] is the concentration of bath gas; W(E+) is the sum of states; and F(E*) is the density of states.39 All the parameters listed can be obtained from quantum theory, except for the collisional efficiency (βc). This variable is generally taken as an empirical parameter with a value between 0 and 1. We find that it is a good approximation to set its value to be 0.1 for hydrocarbon radical systems.40-42 Actually, the rate constants increase less than a factor of 3 when increasing from 0.1 to 1. In this work, βc was taken as 0.1 and kept constant for all the calculations. The sum of states W(E+) and the density of states F(E*) are calculated using the BeyerSwinehart algorithm.43 Hindered rotor effects are ignored because they changed the rate constant by less than a few percent.
Table 1. Comparison of the Calculated Geometry of 1-Choroethane, 1-Chloroethyl, and the 1-Chloroethyl Decomposition Transition State Structurea
R (C1C2) R (Cl7C2) R (H6C2) R (H4C1) ∠ Cl7C2C1 ∠ H6C2C1 ∠ H4C1C2 ∠ Cl7C2C1H4 ∠ H6C2C1H5
1-chloroethyl
transition state
1-chloroethene
1.484 1.717 1.082 1.093 118.56 122.32 109.68 171.10 204.53
1.332 1.722 1.082 1.844 122.97 123.47 107.36 88.69 9.93
1.331 1.730 1.083 123.22 123.52 0.10
a
Structures are optimized at MP2/6-31G* level. Units are Å for bond lengths and degrees for angles.
2.3. Canonical Transition-State Theory (CTST). Under high-pressure-limit conditions, the unimolecular rate constant does not depend on pressure and the expression simplifies to canonical transition-state theory (CTST),
kT Q+ exp(-E0/RT) h Q
k∞ ) L
(3)
where Q and Q+ are the complete vibrational-rotational partition functions for the reactant and the transition state. 3. Results and Discussions Table 1 shows the structures of 1-chloroethyl radical, the transition state of the 1-chloroethyl radical decomposition reaction, and 1-chloroethene product calculated at the MP2/ 6-31G* level. For the 1-chloroethyl radical, carbon C2 has one unpaired electron, which is the reason for the high activity of this radical. The C1 and C2 bond length is 1.484 Å and is slightly shorter than that of 1-chloroethane, indicating a stronger C-C bond compared with that of 1-chloroethane. As the decomposition reaction occurs, the H4-C1 bond length increases from 1.082 Å for the 1-chloroethyl radical and reaches 1.844 Å at the transition state, revealing this bond-rupture mode. The C1-C2 bond length decreases and reaches 1.332 Å at the transition state, which is very close to the equilibrium bond length of 1-chloroethene, 1.331 Å. Meanwhile, the C1-C2 structure becomes mostly planar, indicating the formation of 1-chloroethene product. The negative frequency corresponding to the bond-rupture mode is 1218 cm-1, a typical value for C-H bond scission. Table 2 lists the calculated energies of the reactant, transition state, and products as well as the heat of reaction and the activation energy for the 1-chloroethyl radical decomposition reaction. The G3 and CBS compound models were chosen because of their proven compromise between accuracy and computational cost for hydrocarbon cracking reactions.44 The calculated reaction pathway is shown in Figure 1. The activation energy of the 1-chloroethyl decomposition reaction is 39.50 kcal/ mol using the G3 method and 38.94 kcal/mol using the CBS method. Compared with the experimental activation energy obtained by Manion and Louw,6 40.14 kcal/mol, the relative error is only 1.59% for the G3 method and 2.99% for the CBS
Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 2983 Table 2. Calculated Energies of 1-Chloroethyl Decomposition Reactiona
G3 CBS experiment a
1-chloroethyl
TS
1-chloroethene
H
heat of reaction
activation energy
-538.5307607 -538.1344765
-538.4678137 -538.0724292
-537.9712769 -537.5759866
-0.501 -0.49995
36.70 36.73
39.50 38.94 40.14b
Units are in Hartrees for the reactant, TS, and products; units are in kcal/mol for heat of reaction and activation energy.
b
Reference 6.
Figure 1. Calculated reaction coordinate of the 1-chloroethyl radical β-scission reaction (structures are optimized at MP2(full)/6-31G* level, and energies listed are calculated using the CBS method).
Figure 2. 1-Chloroethyl radical decomposition reaction rate results as a function of pressure at T ) 900 K.
method. This proves that both composite energy methods can accurately predict reaction energetics. The heat of reaction is 36.70 kcal/mol using the G3 method and 36.73 kcal/mol using the CBS method. For unimolecular reactions, rate constants can be dependent on pressure. The pressure effect of this reaction rate is illustrated in Figure 2, and this type of plot is usually referred to as an S-curve. The calculations were performed at a temperature of 900 K using the CBS compound model. 900 K was chosen because it is in the middle of the experimental temperature range.5 According to the rate dependence on pressure, three different regions can be classified in this figure. The right-hand region is referred to as the high-pressure-limit region, where the reaction rate constant does not depend on pressure. CTST is implemented in this region to obtain the theoretical reaction rate. The middle region is referred to as the falloff region, where the reaction rate strongly depends on pressure. RRKM theory is implemented in this region to obtain the theoretical reaction rate. The left-hand region is referred to as the low-pressurelimit region, where the reaction rate is so slow that it does not have any practical applications. Therefore, this region is not the subject of this research.
Figure 3. (a-c) RRKM theory reaction rates for the 1-chloroethyl decomposition reaction using different bath gases compared with experimental data from Knyazev et al.5
In the falloff region, the reaction rate constant is calculated by applying RRKM theory with G3 and CBS composite energy methods. The calculated results, together with the available experimental data,5 are shown in Figure 3, parts a-c. In this work, the collisional efficiency (βc) in the RRKM expression is taken as 0.1 and maintained constant for all calculations. Three bath gasessHe, N2, and Arswith different concentrations are considered. The bath gas influences the reaction rate through the Lennard-Jones collision frequency term (ZLJ). It can be seen from Figure 3 that the G3 and CBS composite energy methods successfully predict the reaction rate under all conditions
2984
Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006
Figure 4. High-pressure canonical transition state theory rate constant results for the 1-chloroethyl decomposition reaction compared with experimental data from Manion and Louw.6
(different bath gases and concentrations) compared with the experimental results. The CBS method estimates a slightly higher reaction rate compared with the G3 method because the activation energy obtained by the G3 method is 0.56 kcal/mol higher than that by CBS. In the high-pressure region, the reaction rate is estimated using CTST and compared with experimental data from Manion and Louw.6 As shown in Figure 4, the G3 and CBS composite energy calculated results show very good agreement with the experiments, while the calculated results using the CBS method are relatively higher compared to the G3 method. The reaction rate constants were then calculated in the pressure range of 0.1-1000 kPa and the temperature range of 600-1000 K using N2 as the bath gas to extend the predictions to a wide range of pressure and temperature conditions. To help chemists and engineers who would like to apply the kinetic calculation results without going into the details of the complicated theories, the kinetic data were regressed using the SAS software program package.45 A linear relationship of log(k) with respect to log(P) and 1/T is introduced. An R2 value of 0.995 indicates a good fit of the resulting model with the quantum chemical calculation results. The obtained kinetic models are shown below.
k (s-1) ) (7.32 × 1010)P0.69 e(-18 727.10/T) when P < P0 (I) k (s-1) ) (3.74 × 1013) e(-20 648.10/T) when P g P0 (II) where P is in units of kPa and T is in units of Kelvin. Model (I) describes the reaction rate in the falloff region, while model (II) applies in the high-pressure region and is derived from the high-pressure-limit CTST. P0 is the switching pressure where the reaction rate changes from the falloff region to the high-pressure-limit region. This pressure was obtained by equalizing models (I) and (II), leading to P0 ) (8.42 × 103) e(-2 782.61/T). At T ) 950 K, the model and the calculated results are shown in Figure 5. This figure indicates the model to be a good description of the complicated quantum-chemical-based chemical kinetic simulation data. The advantage of this simple model is that it can be easily applied, even under conditions where pressure is a factor, meaning it has broad applications. 4. Conclusions In this work, the kinetic modeling of the 1-chloroethyl radical decomposition reaction thermodynamics and kinetics were performed using Gaussian-3 (G3) and complete basis set (CBS)
Figure 5. 1-Chloroethyl radical decomposition reaction kinetic models at T ) 950 K.
composite energy methods. The reaction energies and kinetic results were in agreement with the limited available experimental data. The G3 and CBS composite energy methods proved to give accurate energetic results for the 1-chloroethyl hydrocarbon cracking reaction. A kinetic model of this reaction with pressure and temperature effects was proposed. For P < P0, k (s-1) ) (7.32 × 1010)P0.69 e(-18 727.10/T); for P g P0, k (s-1) ) (3.74 × 1013) e(-20 648.10/T), where P is in units of kPa, T is in units of Kelvin, and switching pressure P0 ) (8.42 × 103) e(-2 782.61/T). The model can be easily applied to different reaction conditions without going into the complicated theoretical details and performing costly quantum chemical calculations. Acknowledgment This work was funded by the State of Arizona through the Office of the Vice President for Research at the University of Arizona. Supercomputer time was provided by the National Computational Science Alliance and used the NCSA HP/Convex Exemplar SPP-2000 at the University of Illinois at UrbanaChampaign. Literature Cited (1) Gardiner, W. C.; Burcat, A. Combustion chemistry; SpringerVerlag: New York, 1984. (2) Warnatz, J.; Gardiner, W. C.; Burcat, A. Combustion Chemistry; Springer-Verlag: New York, 1984. (3) Chang, W. D.; Senkan, S. M. Detailed Chemical Kinetic Modeling Of Fuel-Rich Trichloroethane/Oxygen/Argon Flames. EnViron. Sci. Technol. 1989, 23 (4), 442-450. (4) Qun, M.; Senkan, S. M. Chemical Kinetic Modeling Of Fuel-Rich Flames Of CH2Cl2/CH4/O2/Ar. Combust. Sci. Technol. 1994, 101 (1-6), 103-134. (5) Knyazev, V. D.; Bencsura, A.; Dubinsky, I. A.; Gutman, D.; Senkan, S. M. The unimolecular decomposition of 1-chloroethyl radical. Symp. (Int.) Combust., [Proc.] 1994, 25, 817-824. (6) Manion, J. A.; Louw, R. Studies In Gas-Phase Hydrogenolysis. 5. Gas-Phase Hydrogenolysis Of ChloroethenesRates, Products, And Computer Modeling. J. Chem. Soc., Perkin Trans. 2 1988, 8, 1547-1555. (7) Lynch, B. J.; Truhlar, D. G. How well can hybrid density functional methods predict transition state geometries and barrier heights? J. Phys. Chem. A 2001, 105 (13), 2936-2941. (8) Jursic, B. S. Ab initio and hybrid density functional theory studies of the forward and reverse barriers for the C2H4+H f C2H5 reaction. J. Chem. Soc., Perkin Trans. 2 1997, 3, 637-641. (9) Truong, T. N.; Truong, T. T. T. A reaction class approach with the integrated molecular orbital plus molecular orbital methodology. Chem. Phys. Lett. 1999, 314 (5-6), 529-533. (10) Truong, T. N. Reaction class transition state theory: Hydrogen abstraction reactions by hydrogen atoms as test cases. J. Chem. Phys. 2000, 113 (12), 4957-4964. (11) Blowers, P.; Masel, R. Engineering energies in introduction approximations for activation hydrogen transfer reactions. AIChE J. 2000, 46 (10), 2041-2052.
Ind. Eng. Chem. Res., Vol. 45, No. 9, 2006 2985 (12) Wong, M. W.; Radom, L. Radical addition to alkenes: Further assessment of theoretical procedures. J. Phys. Chem. A 1998, 102 (12), 2237-2245. (13) Wong, M. W.; Radom, L. Radical-Addition to AlkenessAn Assessment of Theoretical Procedures. J. Phys. Chem. 1995, 99 (21), 85828588. (14) Wong, M. W.; Pross, A.; Radom, L. Comparison of the Addition of CH3•, CH20H•, and CH2CN• Radicals to Substituted Alkenes: A Theoretical Study of the Reaction Mechanism. J. Am. Chem. Soc. 1994, 116 (14), 6284-6292. (15) Xiao, Y. T.; Longo, J. M.; Hieshima, G. B.; Hill, R. J. Understanding the kinetics and mechanisms of hydrocarbon thermal cracking: An ab initio approach. Ind. Eng. Chem. Res. 1997, 36 (10), 4033-4040. (16) Saeys, M.; Reyniers, M. F.; Marin, G. B.; van Speybroeck, V.; Waroquier, M. Ab initio calculations for hydrocarbons: Enthalpy of formation, transition state geometry, and activation energy for radical reactions. J. Phys. Chem. A 2003, 107 (43), 9147-9159. (17) Hehre, W. J. Ab initio molecular orbital theory; Wiley: New York, 1986. (18) Jensen, F. An introduction to computational chemistry; Wiley: New York, 1998. (19) Møller, C.; Plesset, M. S. Note on an approximation treatment for many-electron systems. Phys. ReV. 1934, 46 (7), 0618-0622. (20) Curtiss, L. A.; Raghavachari, K.; Pople, J. A. Gaussian-2 Theorys Use of Higher-Level Correlation Methods, Quadratic ConfigurationInteraction Geometries, and 2nd-Order Møller-Plesset Zero-Point Energies. J. Chem. Phys. 1995, 103 (10), 4192-4200. (21) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Bassolov, V.; Pople, J. A. Gaussian-3 theory using reduced Møller-Plesset order. J. Chem. Phys. 1997, 110, 4703. (22) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Rassolov, V.; Pople, J. A. Gaussian-3 (G3) theory for molecules containing first and second-row atoms. J. Chem. Phys. 1998, 109 (18), 7764-7776. (23) Baboul, A. G.; Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-3 theory using density functional geometries and zero-point energies. J. Chem. Phys. 1999, 110 (16), 7650-7657. (24) Kedziora, G. S.; Pople, J. A.; Rassolov, V. A.; Ratner, M. A.; Redfern, P. C.; Curtiss, L. A. The relativistic Dirac-Coulomb-Fock effect on atomization energies. J. Chem. Phys. 1999, 110 (15), 7123-7126. (25) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Allaham, M. A.; Shirley, W. A.; Mantzaris, J. A Complete Basis Set Model Chemistry. 1. The Total Energies of Closed-Shell Atoms and Hydrides of the 1st-Row Elements. J. Chem. Phys. 1988, 89 (4), 2193-2218. (26) Montgomery, J. A.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A complete basis set model chemistry. VII. Use of the minimum population localization method. J. Chem. Phys. 2000, 112 (15), 65326542. (27) Morihovitis, T.; Schiesser, C. H.; Skidmore, M. A. An ab initio study of beta-fragmentation reactions in some alkoxyacyl (alkoxycarbonyl) and related radicals. J. Chem. Soc., Perkin Trans. 2 1999, 10, 2041-2047. (28) Montgomery, J. A.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A complete basis set model chemistry. VI. Use of density functional geometries and frequencies. J. Chem. Phys. 1999, 110 (6), 2822-2827. (29) Mayer, P. M.; Parkinson, C. J.; Smith, D. M.; Radom, L. An assessment of theoretical procedures for the calculation of reliable free radical thermochemistry: A recommended new procedure. J. Chem. Phys. 1998, 108 (2), 604-615. (30) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A. A complete basis set model chemistry. 5. Extensions to six or more heavy atoms. J. Chem. Phys. 1996, 104 (7), 2598-2619.
(31) Montgomery, J. A.; Ochterski, J. W.; Petersson, G. A. A Complete Basis Set Model Chemistry. 4. An Improved Atomic Pair Natural Orbital Method. J. Chem. Phys. 1994, 101 (7), 5900-5909. (32) Petersson, G. A.; Allaham, M. A. A Complete Basis Set Model Chemistry. 2. Open-Shell Systems and the Total Energies of the 1st-Row Atoms. J. Chem. Phys. 1991, 94 (9), 6081-6090. (33) Petersson, G. A.; Tensfeldt, T. G.; Montgomery, J. A. A Complete Basis Set Model Chemistry. 3. The Complete Basis Set-Quadratic Configuration-Interaction Family of Methods. J. Chem. Phys. 1991, 94 (9), 6091-6101. (34) Zheng, X.; Blowers, P. The investigation of hydrocarbon cracking reaction energetics using composite energy method. Mol. Simul. 2005, in press. (35) 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.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; 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.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98, revision A.7; Gaussian, Inc.: Pittsburgh, PA, 1998. (36) Gonzalez, C.; Schlegel, H. B. An Improved Algorithm For ReactionPath Following. J. Chem. Phys. 1989, 90 (4), 2154-2161. (37) Scott, A. P.; Radom, L. Harmonic vibrational frequencies: An evaluation of Hartree-Fock, Møller-Plesset, quadratic configuration interaction, density functional theory, and semiempirical scale factors. J. Phys. Chem. 1996, 100 (41), 16502-16513. (38) Gilbert, R. G.; Smith, S. C. Theory of unimolecular and recombination reactions; Blackwell Scientific Publications: Oxford, England, 1990. (39) Holbrook, K. A.; Pilling, M. J.; Robertson, S. H. Unimolecular Reactions; John Wiley & Sons: Chichester, New York, 1996. (40) Zheng, X.; Blowers, P.; Zhang, N. Application of compound models for estimating rate constants of hydrocarbon thermal cracking reactions: The neopentyl radical beta-scission reaction. Mol. Simul. 2005, 31 (9), 615621. (41) Zheng, X.; Blowers, P. Kinetic Modeling of the Propyl Radical β-Scission Reaction: An Application of Composite Energy Methods. Ind. Eng. Chem. Res. 2006, 45 (2), 530-535. (42) Zheng, X.; Blowers, P. The application of composite energy methods to n-butyl radical β-scission reaction kinetic estimations. Theor. Chem. Acc. 2005, in press. (43) Beyer, T.; Swinehart, D. F. Number of Multiply-Restricted Partitions. Commun. ACM 1973, 16 (6), 379. (44) Zheng, X.; Blowers, P. In Quantum chemical prediction of hydrocarbon cracking reactions; Proceedings of AIChE Annual Meeting, Austin, TX, 2004; American Institute of Chemical Engineers: Austin, TX, 2004; p 57e. (45) SAS Institute, version 8. SAS Institute: Cary, NC, 1999.
ReceiVed for reView December 16, 2005 ReVised manuscript receiVed March 6, 2006 Accepted March 13, 2006 IE051403G
