基元反应N（～4S）+O_2（X～3∑_g～-）→NO（X～2Π）+O（～3P）的准经典轨线研究与辛算法计算
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摘要
基于Sayós等通过ab initio电子结构计算数据拟合得到的新的基态势能面、采用四阶显式辛格式【55】,我们对基元气相反应N(4S)+O2(X3Σg )→NO(X2Π)+O(3P)进行了深入的准经典轨线研究。我们的工作主要由两个方面组成:
第一方面工作是对分子反应动力学的准经典轨线研究中的数值积分方法的改进。我们基于Sayós 等提供的新的基态势能面,采用四阶显式辛格式计算了上述气相反应的准经典轨线,并与常用的四阶Runge-Kutta 法的计算结果进行了比较。结果显示,四阶显式辛格式保持了反应体系的能量守恒,在分子反应动力学的准经典轨线计算中优于四阶Runge-Kutta 法。
第二方面工作是采用Sayós 等提供的更合理的势能面来研究该气相反应。我们采用Sayós 等提供的新的基态势能面,并用四阶显式辛格式替代常用的四阶Runge-Kutta 法,对上述气相反应进行了系统的准经典轨线研究。这方面工作可大体分为两个部分。首先,我们详细分析了相对平动能、O2 分子的振动能级和转动能级对反应几率和反应截面的影响,并与Gilibert等的工作进行了比较,结果发现,我们的计算值与实际符合得更好。第二,我们计算了上述气相反应在T=300、500、700K 时选态反应速率常数和热速率常数,并将热速率常数的计算值与以前的理论和实验值比较。结果显示,我们计算的热速率常数与Baulch 等推荐的实验值和Sayós 等的变分过渡态理论值都符合得很好,并且明显比Gilibert等【39】的准经典轨线计算值合理。
The molecular reaction dynamics investigates the microscopicdynamics and the mechanism of chemical reaction in the level of atomand molecule, and it is a foundation of the macroscopic chemicaldynamics. Because of the molecular reaction dynamics study, ourknowledge about the chemical reaction has undergone an improvementfrom macroscopic to microscopic, from qualitative to quantitative,from static to dynamic. The study of molecular reaction dynamicsenables researchers to actually deepen the comprehending of chemicalreaction and reveal the elementary rule of chemical reaction in bothexperiment and theory.
In 1960s, it is the application of crossed-molecule beamstechnique in the experimental study of chemical reaction that enables
researchers to observe the chemical reaction in the level of atom andmolecule. In 1970s, the study of chemical reaction has been advancedto the level of the quantum state-to-state reaction because of theapplication of laser technique in the laboratory. Thus, the molecularreaction dynamics becomes a new branch of chemical reaction study inthe chemical physics area. Recently, some precise experimentaltechnique such as crossed-molecule beams, laser induced fluorescenceand chemical laser and so on, have been widely applied in the study ofmolecular reaction dynamics. Especially, with the appearing of thefemto-second laser, researchers are capable of investigating theinteraction of inter-molecule and inner-molecule in a smaller anddeeper level.
From the theoretical point of view, the molecular reactiondynamics study has the classical method, the classical-quantummethod and the quantum method. The quantum method, which baseson the quantum mechanical principle, can actually describe the motionof nuclei in the electronic potential energy surface and provide someaccurate results of reaction dynamics study. However, the classicalmethod (that is the quasiclassical trajectory method here) is the mostwidely applied method of them. This method solves the trajectory ofnuclei motion in the electronic potential energy surface with theclassical mechanics, and then all kinds of macroscopic andmicroscopic reaction dynamical results can be obtained by thestandard statistical average. Its physical image is explicit and itsmathematic management is simple, especially, the classical method isnot enslaved to the dimension of reaction system and can dealt withthe more complex reaction system.
At the present time, the experimental and the theoretical study ofmolecular reaction dynamics are very active. To obtain the accurate
electronic potential energy surface is the first important task, thus, weperform a brief introduction about the ab initio electronic structurecalculation and the experiential LEPS in chapter 2. The quasiclassicaltrajectory method, which is composed of the canonical equations ofreaction system, the selecting of initial conditions and the evaluationof dynamical variables and so on, has also described in detail in thischapter. In the quasiclassical trajectory (QCT) method, motion of atomicnuclei on the electronic potential energy surface (PES) has beenexpressed by the canonical equations of Hamiltonian system and thenumerical integration method frequently adopted is Runge-Kutta orRunge-Kutta-Gear scheme. The improvement of the QCT calculatedresults of molecular reaction dynamics depends greatly on the PESaccuracy of the reaction system, and the numerical integration methodis rarely noticed. However, numerical integration methods, especiallythose conserving constancies of the reaction system, are alsoadvantageous in the QCT study. The Hamiltonian system has thesymplectic structure. In the early 1980s, Ruth and Feng Kangrespectively advanced the symplectic algorithm that is a differencemethod preserving the symplectic structure of Hamiltonian system,and then Feng Kang, Qin Mengzhao and Yoshida et al. carried out asystemic study on the symplectic algorithm. Up to the present, thesymplectic algorithm has been widely applied to astronomy, plasmaphysics, quantum mechanics and other fields. We have introduced thesymplectic algebra and Hamiltonian system in chapter 3. In the chapter,we primarily introduce the canonical equations of Hamiltonian system,symplectic algebra and some available symplectic algorithms. The atom and diatomic molecule reaction system whose numberof atomic nuclei considered is smaller, has been widely investigated in
the molecular reaction dynamics. Moreover, the accurate PES of thereaction system can be obtained by ab initio electronic structurecalculation, and researchers have accumulated an abundant experimentdata. The elementary atmospheric reaction researched in this paper,N(4S)+O2(X3Σg )→NO(X2Π)+O(3P), ?rH0 K =-32.09kcal/mol -0 (1)and its reverse reaction play an important role in the Earth’satmospheric chemistry and combustion processes. Because it isexoergic by 1.39eV, vibrational levels of NO product molecule up tov=7 quanta become populated. This reaction is important as a sourceof infrared chemiluminescence from the upper atmosphere, since theNO product molecule radiates via the fundamental or the first overtonebands. The kinetics of NO formation is of interest in the context ofshock heat air, supersonic expansion of exhaust gases and combustionprocesses with hydrocarbon-air mixture. High temperature studies ofthe kinetics and dynamics of reaction (1) and its reverse one are alsosignificant to interpret the chemical and physical phenomena takingplace during the re-entry of spacecrafts into the Earth’s atmosphere. Recently, a large number of ab initio studies have been presentedabout the ground 2 A′and the first excited 4A′PES involved inreaction (1). In 2002, two ab initio studies on both PESs have beenperformed by Sayós et al. by means of the complete active spaceself-consistent field, second-order perturbation theory over theCASSCF wave function, MR-CI calculations and density functionalmethods covering a wide range of configurational arrangements. Whatis more, Sayós et al. presented analytical PESs for both 2A′and 4A′states based on their accurate ab initio data along with theircomplemented CASPT2 points by the MBE formalism. To betterexplain the exoergicity of reaction (1) and the correspondingexperimental rate constant at 300 K, some experimental data were also
introduced in their fits. Their analytical 2A′PES exhibits severalstationary points that have not been introduced in the previousanalytical surfaces, and depicts accurately the NO2 (X2A1) minimum. Using the symplectic algorithm and the new ground PES by abinitio electronic structure calculation carried by Sayós et al., the QCTstudy is essential in the dynamical study of reaction (1). Therefore,basing on the new ground PES provided by Sayós et al., we havepresented the computation of quasiclassical trajectories for reaction (1)with the explicit fourth order symplectic algorithm (S4) that conservesthe energy of reaction system in chapter 4, and the computed results ofthe fourth order Runge-Kutta scheme (RK4) are also compared. It is known in chapter 4 that RK4 cannot preserve symplecticstructure and energy conservation of the reaction system, which resultin the bad veracity of the trajectory calculation. Because of the loss oftotal energy, RK4 cannot rightly reflect both the colliding mode andthe reaction mode of the trajectories. The amplitude of vibration of thereactant O2 molecule becomes gradually small with the time increasing,and the initial condition of O2 molecule is correspondingly trans-formed before the reaction. Since RK4 also reduce the amplitude ofthe product NO molecule, it cannot guarantee the accuracy of the studyof the rotation-vibrational level distribution of the product molecule.However, S4 maintains symplectic structure and energy conservationof the reaction system and can actually describe the collidingtrajectories of the reaction system. Moreover, since S4 can chooselarger time step size, it may markedly save the computed time of theQCT study. It is concluded that S4 is better than RK4 in the QCT studyof the molecular reaction reaction. By the comparison of theoretical and experimental data, the newground PES by fitting ab initio electronic structure calculating data
第一方面工作是对分子反应动力学的准经典轨线研究中的数值积分方法的改进。我们基于Sayós 等提供的新的基态势能面,采用四阶显式辛格式计算了上述气相反应的准经典轨线,并与常用的四阶Runge-Kutta 法的计算结果进行了比较。结果显示,四阶显式辛格式保持了反应体系的能量守恒,在分子反应动力学的准经典轨线计算中优于四阶Runge-Kutta 法。
第二方面工作是采用Sayós 等提供的更合理的势能面来研究该气相反应。我们采用Sayós 等提供的新的基态势能面,并用四阶显式辛格式替代常用的四阶Runge-Kutta 法,对上述气相反应进行了系统的准经典轨线研究。这方面工作可大体分为两个部分。首先,我们详细分析了相对平动能、O2 分子的振动能级和转动能级对反应几率和反应截面的影响,并与Gilibert等的工作进行了比较,结果发现,我们的计算值与实际符合得更好。第二,我们计算了上述气相反应在T=300、500、700K 时选态反应速率常数和热速率常数,并将热速率常数的计算值与以前的理论和实验值比较。结果显示,我们计算的热速率常数与Baulch 等推荐的实验值和Sayós 等的变分过渡态理论值都符合得很好,并且明显比Gilibert等【39】的准经典轨线计算值合理。
The molecular reaction dynamics investigates the microscopicdynamics and the mechanism of chemical reaction in the level of atomand molecule, and it is a foundation of the macroscopic chemicaldynamics. Because of the molecular reaction dynamics study, ourknowledge about the chemical reaction has undergone an improvementfrom macroscopic to microscopic, from qualitative to quantitative,from static to dynamic. The study of molecular reaction dynamicsenables researchers to actually deepen the comprehending of chemicalreaction and reveal the elementary rule of chemical reaction in bothexperiment and theory.
In 1960s, it is the application of crossed-molecule beamstechnique in the experimental study of chemical reaction that enables
researchers to observe the chemical reaction in the level of atom andmolecule. In 1970s, the study of chemical reaction has been advancedto the level of the quantum state-to-state reaction because of theapplication of laser technique in the laboratory. Thus, the molecularreaction dynamics becomes a new branch of chemical reaction study inthe chemical physics area. Recently, some precise experimentaltechnique such as crossed-molecule beams, laser induced fluorescenceand chemical laser and so on, have been widely applied in the study ofmolecular reaction dynamics. Especially, with the appearing of thefemto-second laser, researchers are capable of investigating theinteraction of inter-molecule and inner-molecule in a smaller anddeeper level.
From the theoretical point of view, the molecular reactiondynamics study has the classical method, the classical-quantummethod and the quantum method. The quantum method, which baseson the quantum mechanical principle, can actually describe the motionof nuclei in the electronic potential energy surface and provide someaccurate results of reaction dynamics study. However, the classicalmethod (that is the quasiclassical trajectory method here) is the mostwidely applied method of them. This method solves the trajectory ofnuclei motion in the electronic potential energy surface with theclassical mechanics, and then all kinds of macroscopic andmicroscopic reaction dynamical results can be obtained by thestandard statistical average. Its physical image is explicit and itsmathematic management is simple, especially, the classical method isnot enslaved to the dimension of reaction system and can dealt withthe more complex reaction system.
At the present time, the experimental and the theoretical study ofmolecular reaction dynamics are very active. To obtain the accurate
electronic potential energy surface is the first important task, thus, weperform a brief introduction about the ab initio electronic structurecalculation and the experiential LEPS in chapter 2. The quasiclassicaltrajectory method, which is composed of the canonical equations ofreaction system, the selecting of initial conditions and the evaluationof dynamical variables and so on, has also described in detail in thischapter. In the quasiclassical trajectory (QCT) method, motion of atomicnuclei on the electronic potential energy surface (PES) has beenexpressed by the canonical equations of Hamiltonian system and thenumerical integration method frequently adopted is Runge-Kutta orRunge-Kutta-Gear scheme. The improvement of the QCT calculatedresults of molecular reaction dynamics depends greatly on the PESaccuracy of the reaction system, and the numerical integration methodis rarely noticed. However, numerical integration methods, especiallythose conserving constancies of the reaction system, are alsoadvantageous in the QCT study. The Hamiltonian system has thesymplectic structure. In the early 1980s, Ruth and Feng Kangrespectively advanced the symplectic algorithm that is a differencemethod preserving the symplectic structure of Hamiltonian system,and then Feng Kang, Qin Mengzhao and Yoshida et al. carried out asystemic study on the symplectic algorithm. Up to the present, thesymplectic algorithm has been widely applied to astronomy, plasmaphysics, quantum mechanics and other fields. We have introduced thesymplectic algebra and Hamiltonian system in chapter 3. In the chapter,we primarily introduce the canonical equations of Hamiltonian system,symplectic algebra and some available symplectic algorithms. The atom and diatomic molecule reaction system whose numberof atomic nuclei considered is smaller, has been widely investigated in
the molecular reaction dynamics. Moreover, the accurate PES of thereaction system can be obtained by ab initio electronic structurecalculation, and researchers have accumulated an abundant experimentdata. The elementary atmospheric reaction researched in this paper,N(4S)+O2(X3Σg )→NO(X2Π)+O(3P), ?rH0 K =-32.09kcal/mol -0 (1)and its reverse reaction play an important role in the Earth’satmospheric chemistry and combustion processes. Because it isexoergic by 1.39eV, vibrational levels of NO product molecule up tov=7 quanta become populated. This reaction is important as a sourceof infrared chemiluminescence from the upper atmosphere, since theNO product molecule radiates via the fundamental or the first overtonebands. The kinetics of NO formation is of interest in the context ofshock heat air, supersonic expansion of exhaust gases and combustionprocesses with hydrocarbon-air mixture. High temperature studies ofthe kinetics and dynamics of reaction (1) and its reverse one are alsosignificant to interpret the chemical and physical phenomena takingplace during the re-entry of spacecrafts into the Earth’s atmosphere. Recently, a large number of ab initio studies have been presentedabout the ground 2 A′and the first excited 4A′PES involved inreaction (1). In 2002, two ab initio studies on both PESs have beenperformed by Sayós et al. by means of the complete active spaceself-consistent field, second-order perturbation theory over theCASSCF wave function, MR-CI calculations and density functionalmethods covering a wide range of configurational arrangements. Whatis more, Sayós et al. presented analytical PESs for both 2A′and 4A′states based on their accurate ab initio data along with theircomplemented CASPT2 points by the MBE formalism. To betterexplain the exoergicity of reaction (1) and the correspondingexperimental rate constant at 300 K, some experimental data were also
introduced in their fits. Their analytical 2A′PES exhibits severalstationary points that have not been introduced in the previousanalytical surfaces, and depicts accurately the NO2 (X2A1) minimum. Using the symplectic algorithm and the new ground PES by abinitio electronic structure calculation carried by Sayós et al., the QCTstudy is essential in the dynamical study of reaction (1). Therefore,basing on the new ground PES provided by Sayós et al., we havepresented the computation of quasiclassical trajectories for reaction (1)with the explicit fourth order symplectic algorithm (S4) that conservesthe energy of reaction system in chapter 4, and the computed results ofthe fourth order Runge-Kutta scheme (RK4) are also compared. It is known in chapter 4 that RK4 cannot preserve symplecticstructure and energy conservation of the reaction system, which resultin the bad veracity of the trajectory calculation. Because of the loss oftotal energy, RK4 cannot rightly reflect both the colliding mode andthe reaction mode of the trajectories. The amplitude of vibration of thereactant O2 molecule becomes gradually small with the time increasing,and the initial condition of O2 molecule is correspondingly trans-formed before the reaction. Since RK4 also reduce the amplitude ofthe product NO molecule, it cannot guarantee the accuracy of the studyof the rotation-vibrational level distribution of the product molecule.However, S4 maintains symplectic structure and energy conservationof the reaction system and can actually describe the collidingtrajectories of the reaction system. Moreover, since S4 can chooselarger time step size, it may markedly save the computed time of theQCT study. It is concluded that S4 is better than RK4 in the QCT studyof the molecular reaction reaction. By the comparison of theoretical and experimental data, the newground PES by fitting ab initio electronic structure calculating data
引文
[1] 赵新生,化学反应理论导论,北京大学出版社,2003,p.1。
[2] M. Karplus, R. Porter, R.D. Sharma, J. Chem. Phys., 49 (1965)3259.
[3] J.D. Doll, Chem.Phys, 3 (1974) 257.
[4] D.G. Truhlar, T. Muckermanin, Physics of Atoms and Molecules, Ed. R.B. Berstein, Plenum, New York, 1979.
[5] S.A. Adelman, J.D. Doll, J. Chem. Phys., 61 (1974) 4242.
[6] S.A. Adelman, J.D. Doll, J. Chem. Phys., 63 (1975) 4906.
[7] S.A. Adelman, J.D. Doll, J. Chem. Phys., 64 (1975) 2375.
[8] S.A. Adelman, Adv. Phys. Chem., 53 (1983) 61.
[9] J.C. Tully, Acc. Chem. Res., 14 (1981) 188.
[10] W.H. Miller, J. Chem. Phys., 54 (1971) 5386.
[11] W.H. Miller, Adv. Chem. Phys., 25 (1974) 69.
[12] W.H. Miller, Adv. Chem. Phys., 30 (1974) 77.
[13] J.D. Doll, Chem. Phys., 3 (1974) 257.
[14] J.D. Doll, J. Chem. Phys., 61 (1975) 954.
[15] R.I. Masel, R.P. Merril, W.H. Miller, J. Chem. Phys., 64(1976)45.
[16] R.I. Masel, R.P. Merril, W.H. Miller, Surf. Sci., 64 (1974) 681.
[17] M.W. Chase, Jr., C.A. Downey, J.R. Downey, D.J Frurip, R.A. McDonald, A.N. Syverud, J. Phys. Chem. Ref. Data, 4 (1985) 1.
[18] T.C. Corcoran, E. J. Beiting, M. O. Mitchell, J. Mol. Spectrosc. 154 (1992) 119.
[19] R. Sayós, C. Oliva, M. González, J. Chem. Phys. 117 (2002) 670.
[20] P. Warneck, Chemistry of the Natural atmosphere, Academic, San Diego, 1998 (Chapter 3).
[21] A. Burcat, G. Dixon-Liwes, M. Frenklach, W. C. Gardiner, R. K. Hanson, S. Salimian, J. Troe, J. Warnatz, R. Zellner, Combustion Chemistry, Ed. W. C. Gardiner, Jr., Springer-Verlag, New York, 1984.
[22] N. Balakrishnan, E. Sergueeva, V. Kharchenko, A. Dalgarno, J. Geo-phys. Res., 105 (2000) 18549.
[23] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 172 (1993) 99.
[24] J.W. Duff, F. Bien, D.E. Paulsen, Geophys. Res. Lett., 21 (1994) 2043.
[25] D. Bose, G.V. Candler, J. Chem. Phys., 107 (1997) 6136.
[26] R. Sayós, J. Hijazo, M. Gilibert, M. González, Chem. Phys. Lett., 284 (1998)101.
[27] P.J.B.S. Caridade, A.J.C. Varandas, J. Phys. Chem. A, 108 (2004) 3556.
[28] G. Suzzi Valli, R. Orrú, E.Clementi, A. Laganá, S.Crocchianti, J. Chem. Phys., 102 (1995) 2825.
[29] R. Sayós, C. Oliva, M. González, J. Chem. Phys., 115 (2001) 1287.
[30] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 172(1993) 99.
[31] A.J.C. Varandas, A.I. Voronin, Mol. Phys., 85 (1995) 497.
[32] S.P. Walch, R.L. Jaffe, J. Chem. Phys., 86 (1987) 6946.
[33] G.S. Valli, R. Orru, E. Clementi, A. Laganá, S. Crocchianti, J. Chem. Phys., 102 (1995) 2825.
[34] D. Reignier, T. Stoecklin, A. Voronin, J. C. Rayez, Phys. Chem. Chem. Phys., 3 (2001) 2726.
[35] M. González, C. Oliva, R. Sayós, J. Chem. Phys., 117 (2002) 680.
[36] J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley, A.J.C. Varandas, Molecular Potential Energy Functions, Wiley, Chichester, 1984.
[37] D.L. Baulch, C.J. Cobos, R.A. Cox, G. Hayman, T. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data, 23 (1994) 873.
[38] U. Kaldor, Chem. Phys. Lett., 185 (1991) 131.
[39] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 178 (1993) 287.
[40] B. Ramachandran, N. Balakrishnan, A. Dalgarno, Chem. Phys. Lett., 332 (2000) 562.
[41] R. Sayós, A. Aguilar, M. Gilibert, M. González, J. Chem. Soc. Faraday Trans., 89 (1993) 3223.
[42] M. Gilibert, X. Giménez, M. González, R. Sayós, A. Aguilar, Chem. Phys., 191 (1995) 1.
[43] N. Balakrishnan, A. Dalgarno, Chem. Phys. Lett., 302 (1999) 485.
[44] P. Defazio, S.K. Gray, C. Petrongolo, C. Oliva, J. Chem. Phys., 115 (2001) 3208.
[45] P. Defazio, C. Petrongolo, C. Oliva, M. González, R. Sayós, J. Chem. Phys., 117 (2002) 3647.
[46] M. Karplus, R.N. Porter, R.D. Sharma, J. Chem. Phys., 43 (1965) 3259.
[47] Ch. Schlier, A. Seiter, J. Phys. Chem. A, 102 (1998) 9399.
[48] K. Feng, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations Computation of Partial Differential Equations, Science Press, Beijing, 1985.
[49] R.D. Ruth, IEEE Trans. Nuc. Sci., 30 (1983) 2669.
[50] K. Feng, J. Comp. Math., 4 (1986) 279.
[51] K.Feng, M.Z.Qin, Progress in Natural Science, 1 (1991) 105.
[52] K. Feng, In Proceedings 1st China-Japan Conference on Numerical Mathematics, World Scientific, Beijing, 1992; Singapore, 1993.
[53] H.Yoshida, Phys.Lett.A, 150 (1990) 262.
[54] S.K. Gray, D.W. Noid, G. Sumpter, J. Chem. Phys., 101 (1994) 4062.
[55] M.Z. Qin, D.L. Wang, M.Q. Zhang, J. Comp. Math., 9 (1991) 211.
[56] H. Yoshida, Phys. Lett. A, 150 (1990) 262.
[57] Ch. Schlier, A. Seiter, Comp. Phys. Comm., 130 (2000) 176.
[58] L.Liu, X.H. Liao, Celest. Mech. Astron., 59 (1994) 221.
[59] B.Gladman, M.Duncan, Candy, J. Celest. Mech. Dynam. Astron., 52 (1991) 221.
[60] X.S. Liu, X.Y. Liu, Z.Y. Zhou, P.Z. Ding, S.F. Pan, Int. J. Quantum Chem., 79 (2000) 343.
[61] X.S. Liu, L.W. Su, X.Y. Liu, P.Z. Ding, Int. J. Quantum Chem., 83 (2001) 303.
[62] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys Chem. Ref. Data, 23 (1994) 847.
[63] 潘毓刚,化学通报,(1980) 197。
[64] B. Liu, J. Chem. Phys., 58 (1973) 1925.
[65] D.G. Siegbahin, B. Liu, J. Chem. Phys., 68 (1973) 2457.
[66] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1950.
[67] V. I. Arnold, Mathematical Method of Classical Mechanics, Springer, Verlag, 1978.
[68] J. E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer, Verlag, 1997.
[69] 秦孟兆,力学与实践,12 (1990) 1.
[70] K. Feng, J. Comput. Math., 4 (1986) 279.
[71] K. Feng, H.M. Wu, M.Z. Qin, J. Comput. Math., 8 (1990) 371.
[72] M.Z. Qin, D.L. Wang, M.Q. Zhang, J. Comp. Math., 9 (1991)211.
[73]石爱民、吴承陨、周忠源、丁培柱,计算物理,15 (1998)153.
[74] J.M. Sanz-Seran, A. Portillo, J. Chem. Phys., 104 (1996) 2349.
[75] K. Feng, H.M. Wu, M.Z. Qin, D.L. Zhang, J. Comput. Math., 7 (1989) 71.
[76] J.M. Sanz-Seran, BIT, 28 (1988) 877.
[77] L.M. Raff, D.L. Thompson, Theory of Chemical Reaction Dynamics, Ed. M. Bear, Florida: CRC Press, Inc. Boca Raton, 1985, (Volume Ⅲ, pp.1).
[78] L.M. Raff, D.L. Thompson, Theory of Chemical Reaction Dynamics, Ed. M. Bear, Florida: CRC Press, Inc. Boca Raton, 1985, (Volume Ⅲ, pp.8).
[79] P. Brumer, M. Karplus, Faraday Disc. Chem. Soc., 55 (1973) 80.
[80] M. González, I. Miquel, R. Sayós, J. Chem. Phys., 115 (2001) 8838.
[81] M. González, I. Miquel, R. Sayós, Chem. Phys. Lett., 360 (2002) 521.
[82] V.I. Shematovich, D.V. Bisikalo, J.C. Gérard, Geophys. Res. Lett., 18 (1991) 1691.
[83] Ya. B. Zeldovich, P. Ya. Sadovnikov, D.A. Frank-Kamenetskii, Oxidation, of Nitrogen in Combustion, Academy of sciences of USSR, Institute of Chemical Physics, Moscow-Leningard, 1947.
[84] G.B. Kisiakowsky, G.G. Volpi, J. Chem. Phys., 27 (1957) 1141.
[85] W.E. Wilson, J. Chem. Phys., 46 (1967) 2017.
[86] A.A. Wesenberg, J.M. Roscoe, N. de Haas, Chem. Phys. Lett., 7 (1970) 597.
[87] M.E. Whitson, Jr., L.A. Darnton, R.J. McNeal, Chem. Phys. Lett., 41 (1976) 552.
[88] A. Rahbee, J.J.Gibson, J. Chem. Phys., 74 (1981) 5143.
[89] R.R. Herm, B.J. Sullivan, M.E. Whitson, Jr., J. Chem. Phys., 79 (1983) 2221.
[90] I.C. Winkler, R.A. Stachnik, J.I. Steinfeld, S.M. Miller, J. Chem. Phys., 85 (1986) 890.
[91] A.J. Barnett, G. Marston, R.P. Wayne, J. Chem. Soc., Faraday Trans., 283 (1987) 1453.
[92] G.E. Caledonia, R.H. Krech, D.B. Oakes, S.J. Lipson, W.A.M. Blumberg, J. Geophys, Res., 105 (2000) 12833.
[2] M. Karplus, R. Porter, R.D. Sharma, J. Chem. Phys., 49 (1965)3259.
[3] J.D. Doll, Chem.Phys, 3 (1974) 257.
[4] D.G. Truhlar, T. Muckermanin, Physics of Atoms and Molecules, Ed. R.B. Berstein, Plenum, New York, 1979.
[5] S.A. Adelman, J.D. Doll, J. Chem. Phys., 61 (1974) 4242.
[6] S.A. Adelman, J.D. Doll, J. Chem. Phys., 63 (1975) 4906.
[7] S.A. Adelman, J.D. Doll, J. Chem. Phys., 64 (1975) 2375.
[8] S.A. Adelman, Adv. Phys. Chem., 53 (1983) 61.
[9] J.C. Tully, Acc. Chem. Res., 14 (1981) 188.
[10] W.H. Miller, J. Chem. Phys., 54 (1971) 5386.
[11] W.H. Miller, Adv. Chem. Phys., 25 (1974) 69.
[12] W.H. Miller, Adv. Chem. Phys., 30 (1974) 77.
[13] J.D. Doll, Chem. Phys., 3 (1974) 257.
[14] J.D. Doll, J. Chem. Phys., 61 (1975) 954.
[15] R.I. Masel, R.P. Merril, W.H. Miller, J. Chem. Phys., 64(1976)45.
[16] R.I. Masel, R.P. Merril, W.H. Miller, Surf. Sci., 64 (1974) 681.
[17] M.W. Chase, Jr., C.A. Downey, J.R. Downey, D.J Frurip, R.A. McDonald, A.N. Syverud, J. Phys. Chem. Ref. Data, 4 (1985) 1.
[18] T.C. Corcoran, E. J. Beiting, M. O. Mitchell, J. Mol. Spectrosc. 154 (1992) 119.
[19] R. Sayós, C. Oliva, M. González, J. Chem. Phys. 117 (2002) 670.
[20] P. Warneck, Chemistry of the Natural atmosphere, Academic, San Diego, 1998 (Chapter 3).
[21] A. Burcat, G. Dixon-Liwes, M. Frenklach, W. C. Gardiner, R. K. Hanson, S. Salimian, J. Troe, J. Warnatz, R. Zellner, Combustion Chemistry, Ed. W. C. Gardiner, Jr., Springer-Verlag, New York, 1984.
[22] N. Balakrishnan, E. Sergueeva, V. Kharchenko, A. Dalgarno, J. Geo-phys. Res., 105 (2000) 18549.
[23] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 172 (1993) 99.
[24] J.W. Duff, F. Bien, D.E. Paulsen, Geophys. Res. Lett., 21 (1994) 2043.
[25] D. Bose, G.V. Candler, J. Chem. Phys., 107 (1997) 6136.
[26] R. Sayós, J. Hijazo, M. Gilibert, M. González, Chem. Phys. Lett., 284 (1998)101.
[27] P.J.B.S. Caridade, A.J.C. Varandas, J. Phys. Chem. A, 108 (2004) 3556.
[28] G. Suzzi Valli, R. Orrú, E.Clementi, A. Laganá, S.Crocchianti, J. Chem. Phys., 102 (1995) 2825.
[29] R. Sayós, C. Oliva, M. González, J. Chem. Phys., 115 (2001) 1287.
[30] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 172(1993) 99.
[31] A.J.C. Varandas, A.I. Voronin, Mol. Phys., 85 (1995) 497.
[32] S.P. Walch, R.L. Jaffe, J. Chem. Phys., 86 (1987) 6946.
[33] G.S. Valli, R. Orru, E. Clementi, A. Laganá, S. Crocchianti, J. Chem. Phys., 102 (1995) 2825.
[34] D. Reignier, T. Stoecklin, A. Voronin, J. C. Rayez, Phys. Chem. Chem. Phys., 3 (2001) 2726.
[35] M. González, C. Oliva, R. Sayós, J. Chem. Phys., 117 (2002) 680.
[36] J.N. Murrell, S. Carter, S.C. Farantos, P. Huxley, A.J.C. Varandas, Molecular Potential Energy Functions, Wiley, Chichester, 1984.
[37] D.L. Baulch, C.J. Cobos, R.A. Cox, G. Hayman, T. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data, 23 (1994) 873.
[38] U. Kaldor, Chem. Phys. Lett., 185 (1991) 131.
[39] M. Gilibert, A. Aguilar, M. González, R. Sayós, Chem. Phys., 178 (1993) 287.
[40] B. Ramachandran, N. Balakrishnan, A. Dalgarno, Chem. Phys. Lett., 332 (2000) 562.
[41] R. Sayós, A. Aguilar, M. Gilibert, M. González, J. Chem. Soc. Faraday Trans., 89 (1993) 3223.
[42] M. Gilibert, X. Giménez, M. González, R. Sayós, A. Aguilar, Chem. Phys., 191 (1995) 1.
[43] N. Balakrishnan, A. Dalgarno, Chem. Phys. Lett., 302 (1999) 485.
[44] P. Defazio, S.K. Gray, C. Petrongolo, C. Oliva, J. Chem. Phys., 115 (2001) 3208.
[45] P. Defazio, C. Petrongolo, C. Oliva, M. González, R. Sayós, J. Chem. Phys., 117 (2002) 3647.
[46] M. Karplus, R.N. Porter, R.D. Sharma, J. Chem. Phys., 43 (1965) 3259.
[47] Ch. Schlier, A. Seiter, J. Phys. Chem. A, 102 (1998) 9399.
[48] K. Feng, Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations Computation of Partial Differential Equations, Science Press, Beijing, 1985.
[49] R.D. Ruth, IEEE Trans. Nuc. Sci., 30 (1983) 2669.
[50] K. Feng, J. Comp. Math., 4 (1986) 279.
[51] K.Feng, M.Z.Qin, Progress in Natural Science, 1 (1991) 105.
[52] K. Feng, In Proceedings 1st China-Japan Conference on Numerical Mathematics, World Scientific, Beijing, 1992; Singapore, 1993.
[53] H.Yoshida, Phys.Lett.A, 150 (1990) 262.
[54] S.K. Gray, D.W. Noid, G. Sumpter, J. Chem. Phys., 101 (1994) 4062.
[55] M.Z. Qin, D.L. Wang, M.Q. Zhang, J. Comp. Math., 9 (1991) 211.
[56] H. Yoshida, Phys. Lett. A, 150 (1990) 262.
[57] Ch. Schlier, A. Seiter, Comp. Phys. Comm., 130 (2000) 176.
[58] L.Liu, X.H. Liao, Celest. Mech. Astron., 59 (1994) 221.
[59] B.Gladman, M.Duncan, Candy, J. Celest. Mech. Dynam. Astron., 52 (1991) 221.
[60] X.S. Liu, X.Y. Liu, Z.Y. Zhou, P.Z. Ding, S.F. Pan, Int. J. Quantum Chem., 79 (2000) 343.
[61] X.S. Liu, L.W. Su, X.Y. Liu, P.Z. Ding, Int. J. Quantum Chem., 83 (2001) 303.
[62] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys Chem. Ref. Data, 23 (1994) 847.
[63] 潘毓刚,化学通报,(1980) 197。
[64] B. Liu, J. Chem. Phys., 58 (1973) 1925.
[65] D.G. Siegbahin, B. Liu, J. Chem. Phys., 68 (1973) 2457.
[66] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, Massachusetts, 1950.
[67] V. I. Arnold, Mathematical Method of Classical Mechanics, Springer, Verlag, 1978.
[68] J. E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer, Verlag, 1997.
[69] 秦孟兆,力学与实践,12 (1990) 1.
[70] K. Feng, J. Comput. Math., 4 (1986) 279.
[71] K. Feng, H.M. Wu, M.Z. Qin, J. Comput. Math., 8 (1990) 371.
[72] M.Z. Qin, D.L. Wang, M.Q. Zhang, J. Comp. Math., 9 (1991)211.
[73]石爱民、吴承陨、周忠源、丁培柱,计算物理,15 (1998)153.
[74] J.M. Sanz-Seran, A. Portillo, J. Chem. Phys., 104 (1996) 2349.
[75] K. Feng, H.M. Wu, M.Z. Qin, D.L. Zhang, J. Comput. Math., 7 (1989) 71.
[76] J.M. Sanz-Seran, BIT, 28 (1988) 877.
[77] L.M. Raff, D.L. Thompson, Theory of Chemical Reaction Dynamics, Ed. M. Bear, Florida: CRC Press, Inc. Boca Raton, 1985, (Volume Ⅲ, pp.1).
[78] L.M. Raff, D.L. Thompson, Theory of Chemical Reaction Dynamics, Ed. M. Bear, Florida: CRC Press, Inc. Boca Raton, 1985, (Volume Ⅲ, pp.8).
[79] P. Brumer, M. Karplus, Faraday Disc. Chem. Soc., 55 (1973) 80.
[80] M. González, I. Miquel, R. Sayós, J. Chem. Phys., 115 (2001) 8838.
[81] M. González, I. Miquel, R. Sayós, Chem. Phys. Lett., 360 (2002) 521.
[82] V.I. Shematovich, D.V. Bisikalo, J.C. Gérard, Geophys. Res. Lett., 18 (1991) 1691.
[83] Ya. B. Zeldovich, P. Ya. Sadovnikov, D.A. Frank-Kamenetskii, Oxidation, of Nitrogen in Combustion, Academy of sciences of USSR, Institute of Chemical Physics, Moscow-Leningard, 1947.
[84] G.B. Kisiakowsky, G.G. Volpi, J. Chem. Phys., 27 (1957) 1141.
[85] W.E. Wilson, J. Chem. Phys., 46 (1967) 2017.
[86] A.A. Wesenberg, J.M. Roscoe, N. de Haas, Chem. Phys. Lett., 7 (1970) 597.
[87] M.E. Whitson, Jr., L.A. Darnton, R.J. McNeal, Chem. Phys. Lett., 41 (1976) 552.
[88] A. Rahbee, J.J.Gibson, J. Chem. Phys., 74 (1981) 5143.
[89] R.R. Herm, B.J. Sullivan, M.E. Whitson, Jr., J. Chem. Phys., 79 (1983) 2221.
[90] I.C. Winkler, R.A. Stachnik, J.I. Steinfeld, S.M. Miller, J. Chem. Phys., 85 (1986) 890.
[91] A.J. Barnett, G. Marston, R.P. Wayne, J. Chem. Soc., Faraday Trans., 283 (1987) 1453.
[92] G.E. Caledonia, R.H. Krech, D.B. Oakes, S.J. Lipson, W.A.M. Blumberg, J. Geophys, Res., 105 (2000) 12833.