激光场中氢分子离子动力学行为的经典理论与保结构计算
摘要
随着强激光技术的飞速发展，目前实验室已可获得峰值强度达到
1020W/cm2的激光脉冲。这极大地推动了激光与物质相互作用的研究，
使得激光与原子、分子、团簇相互作用的研究成为基础理论研究中极
为活跃的前沿课题。在实验上，强激光技术的引入，使人们有可能从
实验上观测到原子或分子在强激光下的瞬态过程，从而导致了现代激
光化学的飞速发展，如飞秒化学。飞秒激光技术的发展，使人们对于
从实验上实时的观测反应过渡态的演化、过渡态的空间结构、化学反
应前后分子之间的矢量关系，从而使深入认识化学反应的规律成为可
能。这些实验研究推动了人们对于原子和分子与强激光脉冲相互作用
的动力学过程的理论研究，使之成为当今理论研究中极为活跃的领域。
在理论和实验研究中，强激光场中分子解离和电离的动力学过程
研究一直是十分活跃的课题之一。虽然激光场中分子动力学的理论研
究是极其复杂的问题，但是有关的实验结果却给出了惊人的规律性，
例如，电离、解离过程中释放出来的动能和解离、电离的碎片的比率
基本上与激光场的性质无关。激光场中分子动力学的理论研究分为两
个层面：一是基于 Born-Oppenheimer 近似条件下的研究，二是非
Born-Oppenheimer 近似条件下的研究。前者在原子核不动的近似条件
下求解电子方程，通过研究电子在激光场中的运动来研究分子的动力
学行为。但是由于分子中的原子核在激光场的作用下也受到力的作用，
I
吉林大学硕士学位论文
也在不断地改变运动状态，且与电子之间有很强的相关，因此，人们
又逐渐在非 Born-Oppenheimer 近似下研究激光场中分子的动力学，进
一步给出了电荷共振增强电离 (CREI)、阈上电离(ATI)和阈上解离
(ATD)等理论解释。
很久以前，人们已经开展了氢分子离子H2 的光解离和光电离问题
+
的实验和理论研究。如处在 1sσg 电子态（吸引态或成键分子态）的氢
分子离子吸收光子后将跃迁到解离态 2pσu(排斥态或反键分子态)。
对于强激光场中原子、分子的动力学过程的理论研究，由于微扰
方法不适用了，数值求解 Coulomb 势与激光场共同作用下原子、分子
的含时 Schr?dinger 方程的方法逐渐受到人们的重视和采用。但是对于
强激光场作用下的多电子原子或多原子分子系统，即便采用目前计算
能力最强的计算机，应用量子力学从头算方法数值求解系统的含时
Schr?dinger 方程也是极其繁难甚至是不可能的。因此人们又重新采用
经典理论来数值研究强激光场中多电子原子或多原子分子的动力学过
程，如高频场中分子的稳定性，电离，Coulomb 爆炸以及原子、小分
子的高次谐波发射等，并得到了令人满意的结果。
采用经典理论研究分子动力学问题，需要数值求解Newton方程或
Hamilton正则方程。过去，数值求解常微分方程常采用Runge-Kutta法
和改进的Runge-Kutta法。1980 年代初， Ruth和冯康基于Hamilton系
统的基本原理：Hamilton正则方程的解的时间演化是辛变换的演化，
提出了Hamilton系统的辛算法。之后，辛算法得到了系统深入地研究，
并广泛应用于许多科学领域。例如，辛算法已经应用于量子力学和强
场物理、天体力学和大气与海洋科学、等离子体和地学等领域的研究
中，并取得了可喜的成果。特别在长时间、多步数的计算中和保持系
统的本质属性上显示出极大的优越性。辛算法在化学反应动力学、分
子动力学的研究中也得到了广泛应用。Leimkuhler指出分子动力学模
拟中使用的振动Newton模型将导致求解一个Hamilton正则方程组，采
用辛算法数值求解是合理的。李延欣等和石爱民等分别报告了将辛算
II
吉林大学硕士学位论文
法应用于A2B模型分子和双原子分子系统的经典轨迹计算，使计算时
间较非辛算法延长了 1-2 个数量级，充分显示了辛算法的优越性。
本文选择了氢分子离子这个最简单的分子，作为研究激光场中分
子动力学行为的研究对象。本文主要开展了以下研究工作：
1）基于经典理论采用经典轨迹方法数值研究了强激光场中 1 维共线氢
分子离子H2 ：采用辛算法数值求解激光场中一维共线模型氢分子离子
+
Hamilton 正则方程的初值问题，计算了经典轨迹；讨论了存活、解离、
电离和Coulomb爆炸等动力学过程的产生机制，利用统计平均方法计
算了存活、解离、电离和Coulomb爆炸等动力学行为的几率演化曲线；
计算和分析了双色场中一维共线模型氢分子离子的经典轨迹和动力学
行为，讨论了与单色场中氢分子离子动力学行为的不同之处。
2）采用了 2 维平面氢分子离子模型，应用经典轨迹方法数值研究了强
激光场中二维平面模型氢分子离子的动力学行为，采用辛算法求得了
大量的经典轨迹，分析了二维模型氢分子离子动力学行为的产生机制，
利用统计平均方法计算了二维模型氢分子离子存活、解离、电离和
Coulomb 爆炸等动力学行为的几率演化曲线，说明了二维模型较一维
模型更接近于激光场中真实的氢分子离子。还计算和研究了引入平行
于二原子核连线方向和垂直于连线方向的两
With the development of the intense laser technology, the short-pulse
laser with the maximun energy about 1020W/cm2 can be obtained in recent
experiments, which prompt the research on the interaction between
materials and intense laser field. Those surprising developments make the
research on the atom, molecule and cluster physics of intense laser field
become the very interesting and excited new topics in the basic theory
research. Because the laser technology is introduced in experiments, it is
possible to observe the transient process of atomic or molecules in an
intense laser pulse, which brings on the development of modern laser
chemistry, for example the femtosecond chemistry. Recently, the
femtosecond laser technology is developed, which makes it possible that
the evolvement of reaction transition-state, the space configuration of
transition-state and the vector relation between the molecules of chemic
reaction front and after are real time observed on experiment and the laws
of chemic reaction are known deeply. These requirements on experiment
promote the theory research of the dynamic process of the interaction of
atoms or molecules in ultrashort intense pulse, and make it become one of
the very active research fields now.
It is still a very active research task that relates the question of
molecular dissociation and ionization in the research of theories and
experiments. Though the molecular dynamics in laser fields is a most
complicated question, the experiment results present the striking law, for
example, the kinetic that is released on the course of ionization and
V
吉林大学硕士学位论文
dissociation and the ratio of the fragments of the ionization and
dissociation are independent of the character of laser fields. There is two
lays for the study of the molecular dynamic in the laser fields: the one of
studies is based on the Born-Oppenheimer approximation condition; the
other is under the non-Born-Oppenheimer approximation. The former
solve the equation of electron under the condition that the nuclei do not
move, and research the molecular dynamic by study the movement of
electrons in the laser fields. But the nuclei get the effect of laser and its
movement is altered continually, even there is very stronger correlation.
So people study the molecular dynamic in the laser field under the
non-Born-Oppenheimer approximation and present the explanation about
the charge-resonance enhanced ionization(CREI), above-threshold
dissociation(ATD), and above-threshold ionization(ATI).
The photodissociation and photoionization of the simplest molecule
H2 has long been the subject of experimental and theoretical study. An
+
H2 molecule in the 1sσg electronic state and a given rovibrational state
+
will absorb photons in making a transition to the final dissociating 2pσu
state(repulsion state or anti-bonding state).
As the laser field is so strong that the traditional perturbation is not
suitable to study the atom-laser and molecule-laser interaction, and the
time-dependent Schr?dinger implies all the physical messages about the
atom and the atom-laser interaction, solving numerically time-dependent
Schr?dinger equation (TDSE) is one of the popular and the reasonable
methods in present theoretical research. In principle, the dynamic
evolution of the atoms and molecules under an intense electromagnetic
field should be described with quantum theory (QT). It is still impossible
to numerically solve the time-dependent Schr?dinger equation for the
molecules or multi-electron atoms under the intense laser fields by
applying the ab initio even if the best computer today is used. So the
classical theory (CT) is used to study the dynamic behaviors of the
molecules and atoms in the intense laser pulse, such as the stabilization of
a molecule in superintense high-frequency laser field, ionization, Coulomb
explosion and the high harmonic generation of atomic or small molecule
in the strong laser f
1020W/cm2的激光脉冲。这极大地推动了激光与物质相互作用的研究，
使得激光与原子、分子、团簇相互作用的研究成为基础理论研究中极
为活跃的前沿课题。在实验上，强激光技术的引入，使人们有可能从
实验上观测到原子或分子在强激光下的瞬态过程，从而导致了现代激
光化学的飞速发展，如飞秒化学。飞秒激光技术的发展，使人们对于
从实验上实时的观测反应过渡态的演化、过渡态的空间结构、化学反
应前后分子之间的矢量关系，从而使深入认识化学反应的规律成为可
能。这些实验研究推动了人们对于原子和分子与强激光脉冲相互作用
的动力学过程的理论研究，使之成为当今理论研究中极为活跃的领域。
在理论和实验研究中，强激光场中分子解离和电离的动力学过程
研究一直是十分活跃的课题之一。虽然激光场中分子动力学的理论研
究是极其复杂的问题，但是有关的实验结果却给出了惊人的规律性，
例如，电离、解离过程中释放出来的动能和解离、电离的碎片的比率
基本上与激光场的性质无关。激光场中分子动力学的理论研究分为两
个层面：一是基于 Born-Oppenheimer 近似条件下的研究，二是非
Born-Oppenheimer 近似条件下的研究。前者在原子核不动的近似条件
下求解电子方程，通过研究电子在激光场中的运动来研究分子的动力
学行为。但是由于分子中的原子核在激光场的作用下也受到力的作用，
I
吉林大学硕士学位论文
也在不断地改变运动状态，且与电子之间有很强的相关，因此，人们
又逐渐在非 Born-Oppenheimer 近似下研究激光场中分子的动力学，进
一步给出了电荷共振增强电离 (CREI)、阈上电离(ATI)和阈上解离
(ATD)等理论解释。
很久以前，人们已经开展了氢分子离子H2 的光解离和光电离问题
+
的实验和理论研究。如处在 1sσg 电子态（吸引态或成键分子态）的氢
分子离子吸收光子后将跃迁到解离态 2pσu(排斥态或反键分子态)。
对于强激光场中原子、分子的动力学过程的理论研究，由于微扰
方法不适用了，数值求解 Coulomb 势与激光场共同作用下原子、分子
的含时 Schr?dinger 方程的方法逐渐受到人们的重视和采用。但是对于
强激光场作用下的多电子原子或多原子分子系统，即便采用目前计算
能力最强的计算机，应用量子力学从头算方法数值求解系统的含时
Schr?dinger 方程也是极其繁难甚至是不可能的。因此人们又重新采用
经典理论来数值研究强激光场中多电子原子或多原子分子的动力学过
程，如高频场中分子的稳定性，电离，Coulomb 爆炸以及原子、小分
子的高次谐波发射等，并得到了令人满意的结果。
采用经典理论研究分子动力学问题，需要数值求解Newton方程或
Hamilton正则方程。过去，数值求解常微分方程常采用Runge-Kutta法
和改进的Runge-Kutta法。1980 年代初， Ruth和冯康基于Hamilton系
统的基本原理：Hamilton正则方程的解的时间演化是辛变换的演化，
提出了Hamilton系统的辛算法。之后，辛算法得到了系统深入地研究，
并广泛应用于许多科学领域。例如，辛算法已经应用于量子力学和强
场物理、天体力学和大气与海洋科学、等离子体和地学等领域的研究
中，并取得了可喜的成果。特别在长时间、多步数的计算中和保持系
统的本质属性上显示出极大的优越性。辛算法在化学反应动力学、分
子动力学的研究中也得到了广泛应用。Leimkuhler指出分子动力学模
拟中使用的振动Newton模型将导致求解一个Hamilton正则方程组，采
用辛算法数值求解是合理的。李延欣等和石爱民等分别报告了将辛算
II
吉林大学硕士学位论文
法应用于A2B模型分子和双原子分子系统的经典轨迹计算，使计算时
间较非辛算法延长了 1-2 个数量级，充分显示了辛算法的优越性。
本文选择了氢分子离子这个最简单的分子，作为研究激光场中分
子动力学行为的研究对象。本文主要开展了以下研究工作：
1）基于经典理论采用经典轨迹方法数值研究了强激光场中 1 维共线氢
分子离子H2 ：采用辛算法数值求解激光场中一维共线模型氢分子离子
+
Hamilton 正则方程的初值问题，计算了经典轨迹；讨论了存活、解离、
电离和Coulomb爆炸等动力学过程的产生机制，利用统计平均方法计
算了存活、解离、电离和Coulomb爆炸等动力学行为的几率演化曲线；
计算和分析了双色场中一维共线模型氢分子离子的经典轨迹和动力学
行为，讨论了与单色场中氢分子离子动力学行为的不同之处。
2）采用了 2 维平面氢分子离子模型，应用经典轨迹方法数值研究了强
激光场中二维平面模型氢分子离子的动力学行为，采用辛算法求得了
大量的经典轨迹，分析了二维模型氢分子离子动力学行为的产生机制，
利用统计平均方法计算了二维模型氢分子离子存活、解离、电离和
Coulomb 爆炸等动力学行为的几率演化曲线，说明了二维模型较一维
模型更接近于激光场中真实的氢分子离子。还计算和研究了引入平行
于二原子核连线方向和垂直于连线方向的两
With the development of the intense laser technology, the short-pulse
laser with the maximun energy about 1020W/cm2 can be obtained in recent
experiments, which prompt the research on the interaction between
materials and intense laser field. Those surprising developments make the
research on the atom, molecule and cluster physics of intense laser field
become the very interesting and excited new topics in the basic theory
research. Because the laser technology is introduced in experiments, it is
possible to observe the transient process of atomic or molecules in an
intense laser pulse, which brings on the development of modern laser
chemistry, for example the femtosecond chemistry. Recently, the
femtosecond laser technology is developed, which makes it possible that
the evolvement of reaction transition-state, the space configuration of
transition-state and the vector relation between the molecules of chemic
reaction front and after are real time observed on experiment and the laws
of chemic reaction are known deeply. These requirements on experiment
promote the theory research of the dynamic process of the interaction of
atoms or molecules in ultrashort intense pulse, and make it become one of
the very active research fields now.
It is still a very active research task that relates the question of
molecular dissociation and ionization in the research of theories and
experiments. Though the molecular dynamics in laser fields is a most
complicated question, the experiment results present the striking law, for
example, the kinetic that is released on the course of ionization and
V
吉林大学硕士学位论文
dissociation and the ratio of the fragments of the ionization and
dissociation are independent of the character of laser fields. There is two
lays for the study of the molecular dynamic in the laser fields: the one of
studies is based on the Born-Oppenheimer approximation condition; the
other is under the non-Born-Oppenheimer approximation. The former
solve the equation of electron under the condition that the nuclei do not
move, and research the molecular dynamic by study the movement of
electrons in the laser fields. But the nuclei get the effect of laser and its
movement is altered continually, even there is very stronger correlation.
So people study the molecular dynamic in the laser field under the
non-Born-Oppenheimer approximation and present the explanation about
the charge-resonance enhanced ionization(CREI), above-threshold
dissociation(ATD), and above-threshold ionization(ATI).
The photodissociation and photoionization of the simplest molecule
H2 has long been the subject of experimental and theoretical study. An
+
H2 molecule in the 1sσg electronic state and a given rovibrational state
+
will absorb photons in making a transition to the final dissociating 2pσu
state(repulsion state or anti-bonding state).
As the laser field is so strong that the traditional perturbation is not
suitable to study the atom-laser and molecule-laser interaction, and the
time-dependent Schr?dinger implies all the physical messages about the
atom and the atom-laser interaction, solving numerically time-dependent
Schr?dinger equation (TDSE) is one of the popular and the reasonable
methods in present theoretical research. In principle, the dynamic
evolution of the atoms and molecules under an intense electromagnetic
field should be described with quantum theory (QT). It is still impossible
to numerically solve the time-dependent Schr?dinger equation for the
molecules or multi-electron atoms under the intense laser fields by
applying the ab initio even if the best computer today is used. So the
classical theory (CT) is used to study the dynamic behaviors of the
molecules and atoms in the intense laser pulse, such as the stabilization of
a molecule in superintense high-frequency laser field, ionization, Coulomb
explosion and the high harmonic generation of atomic or small molecule
in the strong laser f
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吉林大学硕士学位论文
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15. Gray S. K., Verosky J. M., Classical Hamiltonian Structures in Wave
Packet Dynamics, J. Chem. Phys., (1994) 100(7), 5011-5022.
16. Gray S. K., Manolopoulos D. E., Symplectic Integrators Tailored to
the Time-dependent Schr?dinger Equation, J. Chem. Phys., (1996)
104(18), 7099-7112.
17. [1] Sanz-Seran J. M., Portillo A., Classical Numerical Integrators for
Wave-Packet Dynamics, J. Chem. Phys., (1996) 104(6), 2349-2355.
58
吉林大学硕士学位论文
[2] Qin Mengzhao. Acta Mathematicae Applicatae Sinica, (1996)
12(3), 284-288.
18. Zhou Z. Y., Ding P. Z., Pan S. P., Study Of a Symplectic Scheme for
the Time Evolution of an Atom in an External Field, J. Korean Phys.
Soc., (1998) 32, 417.
19. Zhou Z. Y., Ding P. Z., Conservation Quantities of the Explicit
Symplectic Scheme for Time-evolution of Quantum System, Chinese
Journal of Atomic and Molecular Physics, (1997) 14(2), 175-184.
20. 丁培柱，李延欣，吴承埙，金明星，量子系统的辛算法，吉林大
学自然科学学报， (1993) 4, 75-78.
21. 吴承埙，丁培柱，陈植，一个模型量子系统的辛差分格式，吉林
大学自然科学学报， (1995) 4, 46-48.
22. 周忠源，吴承埙，丁培柱，关于波函数模方逐步归一化修正的注
记，计算物理， (1997) 14(4-5), 431-432.
23. 吴承埙，丁培柱，陈植，一个强场模型的哈密顿形式，计算物理，
(1996) 13(4), 501-504.
24. [1]Qin Mengzhao. Acta Mathematicae Applicatae Sinica,(1996),12(3),
284-288. [2] J. M. Sanz-Serna and A. Portillo, J. Chem.
Phys.(1996),104(6), 2349. [3] 石爱民，吴承埙，周忠源，丁培柱，
时间相关外场中量子系统时间演化的辛格式，计算物理， (1998)
15(2), 153-158.
25. 吴承埙，周忠源，丁培柱，可分线性哈密顿系统显式辛格式的守
恒量，计算物理， (1997) 14(4-5), 702-704.
26. ladman B., Duncan M., Candy J., Symplectic Integrators for
Long-term Integrations in Celestial Mechanics, Celestial Mech.
Dynam. Astronomy, (1991) 52, 221-240.
27. [1] 季江徽，廖新浩，刘林，辛差分格式的守恒量及其稳定性，
计算物理， (1997) 14(1), 68-74. [2] 廖新浩，刘林，辛算法在限
59
吉林大学硕士学位论文
制性三体问题数值研究中的应用，计算物理， (1995) 12(1),
102-108.
28. Cary J. R., Doxas I., An Explicit Symplectic Integration Scheme for
Plasma Simulations, J. Compu. Phys., (1993) 107, 98-104.
29. Kol A., Laird B. B., Leimkuhler B. J., A Symplectic Method for
Rigid-body Molecular Simulation, J. Chem. Phys., (1997) 107,
2580-2588.
30. Cao Y., Liu J. B., Yang K. Q., Liu H., Li Y. M., Hamilton Description
of Seismic Wave Propagation and Application of Geometric
Quantization. “ 保 结 构 算 法 国 际 研 讨 班 ”， CCAST-WL
WORKSHOP: Vol. 6(1), International Workshop on
Structure-Preserving Algorithm, held at China Center of Advanced
Science and Technology, Beijing, P. R. China, March (2001) 25-31,
43-58.
31. hu W. S., Zhao X. S., Tang Y. Q., New Time-dependent Methods in
Quantum Scattering, J. Chem. Phys., (1996) 104(6), 2271-2274.
32. Zhu W. S., Zhao X. S., Tang Y. Q., Numerical Methods With a High
Order of Accuracy Applied in the Quantum System, J. Chem. Phys.,
(1996) 104(6), 2275-2286.
33. Leimkuhler B. J., Skeel R., Symplectic Numerical Integrators in
Constrained Hamiltonian System, J. Comput. Phys., (1994) 112,
117-125.
34. Bond S. D., Leimkuhler B. J., Laird B. B., The Nosé-Poincaré
Method for Constant Temperature Molecular Dynamics, J. Comput.
Phys., (1999) 151(1), 114-134.
35. Sturgeon J. B., Laird B. B., Symplectic Algorithm for
Constant-pressure Molecular Dynamic Using a Nosé-Poincaré
Thermostat, J. Chem. Phys., (2000) 12(8), 3474-3482.
60
吉林大学硕士学位论文
36. 李延欣，丁培柱，吴承埙，金明星，A2B 模型分子经典轨迹的辛
算法计算，高等学校化学学报， (1994) 15(8), 1181-1186.
37. 石爱民，母英魁，丁培柱，N2双原子系统经典轨迹的辛算法计算，
计算物理， (1997) 14(4-5), 433-434.
38. 石爱民，衣汉威，王毅，丁培柱，异核双原子分子经典轨迹的辛
格式计算及与 R-K 法的比较，原子分子物理学报，（1998）7 月增
刊，101-103.
39. Szczepan Chelkowski, Tao Zuo, Osman Atabek, and André D.
Bandrauk, Dissociation, ionization, and Coulomb explosion of H2 in+
an intense laser field by numerical integration of the time dependent
Schr?dinger equation, Phys.Rev.A., 1995, 52(4), 2977-2983.
40. (1)赵新生，《化学反应理论导论》，北京大学出版社，2003，第一
版，23-39. (2) M.Kapplus and L.M. Raff, Theoretical Investigations
of Reactive Collosions in Molecular Beams:K+CH3I, JCP , (1966),
41(5),1267-1277.
41. 秦孟兆，辛几何及计算哈密顿力学，力学与实践， (1990) 12(2)
1-20.
42. Feng K., Difference Schemes for Hamiltonian Formalism and
Symplectic Geometry, J.Comput.Math., (1986) 4 (3), 279-289.
43. Feng K., Wu H. M., Qin M. Z., Symplectic Schemes for Linear
Hamiltonian Canonical System, J.Comput.Math., (1990) 8 (4),
371-380.
44. Qin M. Z., Wang D. L., Zhang M. Q., Explicit Symplectic Difference
Schemes for Separable Hamiltonian System, J. Comp., Math., (1991)
9(3), 211-221.
45. J T Lin ,T L Lai ,D S Chuu and T F Jiang ,Quantum dynamics of a
diatomic molecule under chirped laser pulse ,J.Phys.B:
At.Mol.Opt.Phys. 31 (1998 ), L117
61
吉林大学硕士学位论文
46. J T Lin and T F Jiang, Cantori barriers in the excitation of a diatomic
molecule by chirped pulses ,J.Phys.B:At.Mol.Opt.Phys. 32
(1999 ) ,4001.
47. 唐敖庆，杨忠志，李前树，《量子化学》，科学出版社，1982，第
一版，10-12.
48. E.Ley-Koo et al,Vibrational Leveles and Franck-Condon Factors of
Diatomic Molecules via Morse Potentialsin a Box,Intern.J.Quantum
Chem. 56, (1995), 175-186.
49. 徐光宪，王祥云，《物质结构》，高等教育出版社，1988，第二版，
113-114.
50. 唐敖庆，杨忠志，李前树，《量子化学》，科学出版社，1982，第
一版，1-1.
51. 褚圣麟，《原子物理学》，高等教育出版社，1979，第一版，252-255.
52. J.G.Leopold and I.C.Percival, Ionisation of highly excited atoms by
electric fields III．Microwave ionisation and excitation, J.Phys.B.
1979, 12(5), 709-721.
53. Yasuhito Ohta, J.Maki, Hidemi Nagao, Kiyoshi Nishikawa, Ionization
process of the hydrogen atom in intense laser
fields:Non-Born-Oppenheimer 1Dmodel calculations, IJQC, 2004,
97,891-895.
54. F.Geccherini, D.Bauer, and F.Cornolti, Harmonic generation by atoms
in circularly polarized two-color laser fields with coplanar
polarizations and commensurate frequencies, Phys.Rev.A., 2003, 68,
053402-1-9
55. John R. Hiskes, Dissociation of Molecular Ions by Electric and
Magnetic Fields, PR., (1961) 122(4), 1207-1217.
2. 何国钟，分子反应动力学如何走向 21 世纪，化学进展，(1994)
6(4)，257-265.
3. S.Chelkowski,A.Conjustean,T.Zuoand A.D.Bandrauk, Dissociative
ionization of H2 + in an intense laser field: Charge-resonance
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interpretation, Phys.Rev.A, 1991,44(7), R4114-R4117.
9. Weixing Qu, Suxing Hu, and Zhizhan Xu, Classical dynamics of H2 +
interacting with an ultrashort intense laser pulse, Phys.Rev.A,
1998,57(3),2219-2222.
10. S.Chelkowski, C.Foisy, and A.D.Bandrauk, electron-nuclear dynamics
of multiphoton H2 dissociatiove ionization in intense laser fields,
+
Phys.Rev.A, 1998,57(2),1176-1185.
11. Yiwu Duan, Wing-Ki Liu, and Jian-Min Yuan, Classical dynamics of
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吉林大学硕士学位论文
ionization, and harmonic generation of a hydrogen molecular ion in
intense laser fields: A collinear model, Phys.Rev.A, 2000, 61, 053403.
12. Ruth R.D., A Canonical Integration Technique, IEEE Trans. Nuclear
Science, (1983) NS-30, 2669-2671.
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Geometry and Differential Equations Computation of Partial
Differential Equations, edited by Feng K., (Science Press, Beijing,
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formalism and symplectic geometry, J.Comp.Math.4(1986)279-289.
[3] K.Feng,H.M. Wu, M.Z. Qin and D.L. Wang, Construction of
canonical difference schemes for Hamiltonian formalism via
generating functions, J.Comp.Math.7(1989)71-96. [4] Z. Shang,
Generating functions for volume –preserving mappings and
Hamilton-Jacobi equations for source-free dynamical systems, Ser.A
37(1994a)1172-1188. [5] G..Sun, Symplectic partitioned Runge-Kutta
methods, J.Comput. Math. 11(1993b)365-372. [6] H.Yoshida,
Construction of higher order symplectic integrators, Phys.lett.A
150(1990)262-268.
14. Ding P. Z., Wu C. X., Mu Y. K., Li Y. X., Jin M. X.,
Square-Preserving and Symplectic Structure and Scheme for
Quantum System, Chinese Phys. Lett., (1996)13(4), 245-248.
15. Gray S. K., Verosky J. M., Classical Hamiltonian Structures in Wave
Packet Dynamics, J. Chem. Phys., (1994) 100(7), 5011-5022.
16. Gray S. K., Manolopoulos D. E., Symplectic Integrators Tailored to
the Time-dependent Schr?dinger Equation, J. Chem. Phys., (1996)
104(18), 7099-7112.
17. [1] Sanz-Seran J. M., Portillo A., Classical Numerical Integrators for
Wave-Packet Dynamics, J. Chem. Phys., (1996) 104(6), 2349-2355.
58
吉林大学硕士学位论文
[2] Qin Mengzhao. Acta Mathematicae Applicatae Sinica, (1996)
12(3), 284-288.
18. Zhou Z. Y., Ding P. Z., Pan S. P., Study Of a Symplectic Scheme for
the Time Evolution of an Atom in an External Field, J. Korean Phys.
Soc., (1998) 32, 417.
19. Zhou Z. Y., Ding P. Z., Conservation Quantities of the Explicit
Symplectic Scheme for Time-evolution of Quantum System, Chinese
Journal of Atomic and Molecular Physics, (1997) 14(2), 175-184.
20. 丁培柱，李延欣，吴承埙，金明星，量子系统的辛算法，吉林大
学自然科学学报， (1993) 4, 75-78.
21. 吴承埙，丁培柱，陈植，一个模型量子系统的辛差分格式，吉林
大学自然科学学报， (1995) 4, 46-48.
22. 周忠源，吴承埙，丁培柱，关于波函数模方逐步归一化修正的注
记，计算物理， (1997) 14(4-5), 431-432.
23. 吴承埙，丁培柱，陈植，一个强场模型的哈密顿形式，计算物理，
(1996) 13(4), 501-504.
24. [1]Qin Mengzhao. Acta Mathematicae Applicatae Sinica,(1996),12(3),
284-288. [2] J. M. Sanz-Serna and A. Portillo, J. Chem.
Phys.(1996),104(6), 2349. [3] 石爱民，吴承埙，周忠源，丁培柱，
时间相关外场中量子系统时间演化的辛格式，计算物理， (1998)
15(2), 153-158.
25. 吴承埙，周忠源，丁培柱，可分线性哈密顿系统显式辛格式的守
恒量，计算物理， (1997) 14(4-5), 702-704.
26. ladman B., Duncan M., Candy J., Symplectic Integrators for
Long-term Integrations in Celestial Mechanics, Celestial Mech.
Dynam. Astronomy, (1991) 52, 221-240.
27. [1] 季江徽，廖新浩，刘林，辛差分格式的守恒量及其稳定性，
计算物理， (1997) 14(1), 68-74. [2] 廖新浩，刘林，辛算法在限
59
吉林大学硕士学位论文
制性三体问题数值研究中的应用，计算物理， (1995) 12(1),
102-108.
28. Cary J. R., Doxas I., An Explicit Symplectic Integration Scheme for
Plasma Simulations, J. Compu. Phys., (1993) 107, 98-104.
29. Kol A., Laird B. B., Leimkuhler B. J., A Symplectic Method for
Rigid-body Molecular Simulation, J. Chem. Phys., (1997) 107,
2580-2588.
30. Cao Y., Liu J. B., Yang K. Q., Liu H., Li Y. M., Hamilton Description
of Seismic Wave Propagation and Application of Geometric
Quantization. “ 保 结 构 算 法 国 际 研 讨 班 ”， CCAST-WL
WORKSHOP: Vol. 6(1), International Workshop on
Structure-Preserving Algorithm, held at China Center of Advanced
Science and Technology, Beijing, P. R. China, March (2001) 25-31,
43-58.
31. hu W. S., Zhao X. S., Tang Y. Q., New Time-dependent Methods in
Quantum Scattering, J. Chem. Phys., (1996) 104(6), 2271-2274.
32. Zhu W. S., Zhao X. S., Tang Y. Q., Numerical Methods With a High
Order of Accuracy Applied in the Quantum System, J. Chem. Phys.,
(1996) 104(6), 2275-2286.
33. Leimkuhler B. J., Skeel R., Symplectic Numerical Integrators in
Constrained Hamiltonian System, J. Comput. Phys., (1994) 112,
117-125.
34. Bond S. D., Leimkuhler B. J., Laird B. B., The Nosé-Poincaré
Method for Constant Temperature Molecular Dynamics, J. Comput.
Phys., (1999) 151(1), 114-134.
35. Sturgeon J. B., Laird B. B., Symplectic Algorithm for
Constant-pressure Molecular Dynamic Using a Nosé-Poincaré
Thermostat, J. Chem. Phys., (2000) 12(8), 3474-3482.
60
吉林大学硕士学位论文
36. 李延欣，丁培柱，吴承埙，金明星，A2B 模型分子经典轨迹的辛
算法计算，高等学校化学学报， (1994) 15(8), 1181-1186.
37. 石爱民，母英魁，丁培柱，N2双原子系统经典轨迹的辛算法计算，
计算物理， (1997) 14(4-5), 433-434.
38. 石爱民，衣汉威，王毅，丁培柱，异核双原子分子经典轨迹的辛
格式计算及与 R-K 法的比较，原子分子物理学报，（1998）7 月增
刊，101-103.
39. Szczepan Chelkowski, Tao Zuo, Osman Atabek, and André D.
Bandrauk, Dissociation, ionization, and Coulomb explosion of H2 in+
an intense laser field by numerical integration of the time dependent
Schr?dinger equation, Phys.Rev.A., 1995, 52(4), 2977-2983.
40. (1)赵新生，《化学反应理论导论》，北京大学出版社，2003，第一
版，23-39. (2) M.Kapplus and L.M. Raff, Theoretical Investigations
of Reactive Collosions in Molecular Beams:K+CH3I, JCP , (1966),
41(5),1267-1277.
41. 秦孟兆，辛几何及计算哈密顿力学，力学与实践， (1990) 12(2)
1-20.
42. Feng K., Difference Schemes for Hamiltonian Formalism and
Symplectic Geometry, J.Comput.Math., (1986) 4 (3), 279-289.
43. Feng K., Wu H. M., Qin M. Z., Symplectic Schemes for Linear
Hamiltonian Canonical System, J.Comput.Math., (1990) 8 (4),
371-380.
44. Qin M. Z., Wang D. L., Zhang M. Q., Explicit Symplectic Difference
Schemes for Separable Hamiltonian System, J. Comp., Math., (1991)
9(3), 211-221.
45. J T Lin ,T L Lai ,D S Chuu and T F Jiang ,Quantum dynamics of a
diatomic molecule under chirped laser pulse ,J.Phys.B:
At.Mol.Opt.Phys. 31 (1998 ), L117
61
吉林大学硕士学位论文
46. J T Lin and T F Jiang, Cantori barriers in the excitation of a diatomic
molecule by chirped pulses ,J.Phys.B:At.Mol.Opt.Phys. 32
(1999 ) ,4001.
47. 唐敖庆，杨忠志，李前树，《量子化学》，科学出版社，1982，第
一版，10-12.
48. E.Ley-Koo et al,Vibrational Leveles and Franck-Condon Factors of
Diatomic Molecules via Morse Potentialsin a Box,Intern.J.Quantum
Chem. 56, (1995), 175-186.
49. 徐光宪，王祥云，《物质结构》，高等教育出版社，1988，第二版，
113-114.
50. 唐敖庆，杨忠志，李前树，《量子化学》，科学出版社，1982，第
一版，1-1.
51. 褚圣麟，《原子物理学》，高等教育出版社，1979，第一版，252-255.
52. J.G.Leopold and I.C.Percival, Ionisation of highly excited atoms by
electric fields III．Microwave ionisation and excitation, J.Phys.B.
1979, 12(5), 709-721.
53. Yasuhito Ohta, J.Maki, Hidemi Nagao, Kiyoshi Nishikawa, Ionization
process of the hydrogen atom in intense laser
fields:Non-Born-Oppenheimer 1Dmodel calculations, IJQC, 2004,
97,891-895.
54. F.Geccherini, D.Bauer, and F.Cornolti, Harmonic generation by atoms
in circularly polarized two-color laser fields with coplanar
polarizations and commensurate frequencies, Phys.Rev.A., 2003, 68,
053402-1-9
55. John R. Hiskes, Dissociation of Molecular Ions by Electric and
Magnetic Fields, PR., (1961) 122(4), 1207-1217.