时域有限差分法在双原子分子光谱量子力学计算中的应用
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摘要
时域有限差分(FDTD)法是在时域中求解电磁场的一种数值计算方法。它把带时间变量的Maxwell微分方程转化为差分方程来求解其电磁场各分量。自从FDTD法创建以来，它主要被应用于研究一般的电磁场和电磁波计算，而用于量子力学计算研究的报导还较少见。
本文首先简要介绍了FDTD法的基本理论和发展，以及一般量子力学计算的常用方法，分析了双原子分子势能函数形式和双原子分子光谱特点。重点阐述了FDTD法应用于双原子分子光谱量子力学的计算，并在对边界作近似截断处理条件下计算了氧分子的基态和激发态的振动光谱。
其次，考虑到其应用范围的推广、物理模型的合理性以及数学逻辑上的严密性，本文以能量守恒为基础对前人提出的近似边界截断处理方法进行了改进，使得FDTD法应用于双原子分子光谱的量子力学计算在理论上更为合理，并将改进后的方法应用于氮分子的基态和激发态振动光谱计算。
最后，将计算所得光谱常数与实验结果进行了分析对比，结果令人满意。表明FDTD法本身及其改进后的截断处理在双原子分子振动光谱的量子力学计算方面是可行的。
FDTD法用于双原子分子量子力学计算不需要使用完备的基函数集合构造波函数，从而避免了因为基函数数目的无限大与实际计算的有限性之间的矛盾。该方法可直接用于时域计算、适于编写通用程序和并行计算，具有简单、直观、易于掌握的特点，从而为量子力学计算在方法上提供了一种新的选择。
Finite-difference time-domain method is a numerical calculation method of finding the solution to the electromagnetic field in the time-domain. It transfers Maxwell Differential Equation with time variant to difference equation so as to find the solution to the electromagnetic field. Since the foundation of FDTD method, it is mainly used in the calculation of the electromagnetic fields and waves. However, there are few reports of the application in the calculation of quantum mechanics.
Firstly, the theories and development of the FDTD method were briefly introduced and the traditi6nal method of quantum calculation was offered. The potential energy functions and the spectrums' characteristic of diatomics were analyzed. The application of FDTD method in vibrational spectrums for diatomics was expatiated, and the vibrational spectrums of ground and excited states for oxygen molecule were calculated with this method by the approximate boundary cut-off .
Secondly, considering rationality of physical model and the strictness of mathematical logic, a novel method to cut-off the boundary was suggested based on the conversation of energy, which makes FDTD more reasonable in the application for the quantum calculation of diatomics spectrums. The improved boundary cut-off method is applied in the calculation of the vibrational spectrums for the ground and excited states of oxygen.
Finally, the spectrum constants were compared with the experimental data and the results are acceptable. Thus it is indicated that FDTD method and the novel boundary cut-off is feasible on the calculating for the vibrational spectrums of diatomics.
FDTD method doesn't need a complete basic set of functions to construct the wave function, therefore the contradiction between the infinity of the number of basic functions and the finity of practical calculation is avoided. FDTD method is feasible for the direct time-domain calculation, the development of
common program and parallel calculation. The method is simple, convenient and easy to be master, and is a new choice for the quantum calculation.
本文首先简要介绍了FDTD法的基本理论和发展，以及一般量子力学计算的常用方法，分析了双原子分子势能函数形式和双原子分子光谱特点。重点阐述了FDTD法应用于双原子分子光谱量子力学的计算，并在对边界作近似截断处理条件下计算了氧分子的基态和激发态的振动光谱。
其次，考虑到其应用范围的推广、物理模型的合理性以及数学逻辑上的严密性，本文以能量守恒为基础对前人提出的近似边界截断处理方法进行了改进，使得FDTD法应用于双原子分子光谱的量子力学计算在理论上更为合理，并将改进后的方法应用于氮分子的基态和激发态振动光谱计算。
最后，将计算所得光谱常数与实验结果进行了分析对比，结果令人满意。表明FDTD法本身及其改进后的截断处理在双原子分子振动光谱的量子力学计算方面是可行的。
FDTD法用于双原子分子量子力学计算不需要使用完备的基函数集合构造波函数，从而避免了因为基函数数目的无限大与实际计算的有限性之间的矛盾。该方法可直接用于时域计算、适于编写通用程序和并行计算，具有简单、直观、易于掌握的特点，从而为量子力学计算在方法上提供了一种新的选择。
Finite-difference time-domain method is a numerical calculation method of finding the solution to the electromagnetic field in the time-domain. It transfers Maxwell Differential Equation with time variant to difference equation so as to find the solution to the electromagnetic field. Since the foundation of FDTD method, it is mainly used in the calculation of the electromagnetic fields and waves. However, there are few reports of the application in the calculation of quantum mechanics.
Firstly, the theories and development of the FDTD method were briefly introduced and the traditi6nal method of quantum calculation was offered. The potential energy functions and the spectrums' characteristic of diatomics were analyzed. The application of FDTD method in vibrational spectrums for diatomics was expatiated, and the vibrational spectrums of ground and excited states for oxygen molecule were calculated with this method by the approximate boundary cut-off .
Secondly, considering rationality of physical model and the strictness of mathematical logic, a novel method to cut-off the boundary was suggested based on the conversation of energy, which makes FDTD more reasonable in the application for the quantum calculation of diatomics spectrums. The improved boundary cut-off method is applied in the calculation of the vibrational spectrums for the ground and excited states of oxygen.
Finally, the spectrum constants were compared with the experimental data and the results are acceptable. Thus it is indicated that FDTD method and the novel boundary cut-off is feasible on the calculating for the vibrational spectrums of diatomics.
FDTD method doesn't need a complete basic set of functions to construct the wave function, therefore the contradiction between the infinity of the number of basic functions and the finity of practical calculation is avoided. FDTD method is feasible for the direct time-domain calculation, the development of
common program and parallel calculation. The method is simple, convenient and easy to be master, and is a new choice for the quantum calculation.
引文
1．何福缄，朱正和．结构化学．人民教育出版社．1979
2．许长存等，原子和分子光谱学，大连理工大学出版社，1989，164
3．曾谨言．量子力学．卷Ⅰ，第二版，北京：科学出版社，1997
4．喀兴林．高等量子力学．北京：高等教育出版社，1999
5. E. Schrdinger, Ann. Physik, 79 (1726), 361, 489; 80 (1926), 437; 81 (1926), 109.
6．朱正和，俞华根．分子结构与分子势能函数．北京：科学出版社，1997．
7. M.Born and J.R.Oppenheimer ,Ann. Physik, 1927, 84: 457.
8. M.Born and K.Huang. Dynamical Theory of Crystal Lattices. Oxford University Press. New York. 1954
9．唐敖庆，量子化学．科学出版社．1982．
10. N. V. Cohan , H. F. Hameka. Born Oppenheimer approximation and the calculation of infrared intensities. J. Chem. Phys. 1966,45:4392
11. M. Mills. Born Oppenheimer failure in sparation of low frequency molecular vibrations. J. Phys. Chem. 1984,88:532
12．徐办庄．分子光谱理论．北京：清化大学出版社，1998．169～174．
13．王国文．原子与分子光谱导论．北京：高等教育出版社，1985．
14. Yee K. S. Numerical solution of initial boundary value problems involving Maxwell' sequations in isotropic media. IEEE Trans. On A P. 1966 , 14:302-307
15．高本庆．《时域有限差分法FDTD Method》．国防工业出版社．1995
16. K. S. Kunz and K. M. Lee. A Three-Dimensional Finite-Differerce Solution of the External Response of an Aircraft to a Complex Transient EM Enviroment:Part 1-The Method and its Implementa Implementation. IEEE Trans. On Electromagnetic compitiblity, EMX-20. 1978, 328-332
17. A. Talove and M. E. Brodwin. Numerical solution of Steadystate Electromagmecic scattering Problems Vsing the Time-dependent Maxwell's Equations. IEEE Trans. Microware Theory Tech. MTT-23. 1975,623-630
18. A. Toflore. Application of the Finite-Difference Time-Domain Method to
Sinu soidal steady-state Electeomagnetic-Pentration Problems. IEEE Trans. On Electromagnetic Compotibility. 1980, EMC-22.191-202
19. K. Umashankar and A. Taflove, A Novel Method to analyzd Electromagnetic scattering of Complex objects. IEEE Trans. On Electromagnetic compitiblity, EMX-24.1981, 397-405
20. Choi D H and Hoefer W J R 1986 IEEE Trans. MTT-34 1464
21．黄整等．时域有限差分法对双原子分子振动光谱的计算的应用．原子分子物理学报．2000，17(4)：577．
22．罗有华等．冷原子在静电势阱中的量子力学效应。物理学报。2002，51(8)：1706-1709
23．凤尔银，郑贤锋．原子与分子物理学报．分立位置表象中双原子分子振动能级的计算．2002，19(3)：341-346
24．刘启能，冯灏．能量自洽法的建立以及对异核双原子分子的应用．原子与分子物理学报．2000，17(2)：339-342
25．刘启能．双原子分子振转能级的计算．四川师范大学学报(自然科学版)．1999．22(6)：730-734
26. R. Kosloff, Time dependent quantum mechanical methods for molecular dynamics. J. Phys. Chem. 1988,92:2087
27. K. P. Huber and G. Herzberg Constants of Diatomic Molecules, Van Nostrand, Princdton ,(1979)
28. J. N. Murrell and K. S. Sorbie ,J. Chem. Soc. Faraday Trans. Ⅱ,70(1974) 1552.
29. Y. P. Varshini ,Rev. Mod. phys.29(1957)664.
30. D. Steele, E. R. Lippincott and J.T . Vandensilce , Rev. Mod. Phys.34(1962)239.
31. I.N. Levine, J. Chem. Phys, 1984, (87):431.
32. Huxley P, Murrell J N. Ground state diatomic potentials. J. Chem. Soc. Faraday Trans. 1983, (79):323
33. Sun. Weiguo, FengHao. An Energy consistent Method for Potential Energy Curves of Diatomic Molecules. J Phys. 1999, B 32:5109
34. P. M. Morse , The potential curves of diatomic molecules. Phys. Rev. 1929,34:57
35. H. M. Hullbert and J. O. Hirschfelder, J. Chem. Phys, 34(1962)239
36. W. N. Whitton and P. J. Kuntz, J. Chem. Phys ,64(1976)3624
37. R. Rydberg, Z. Phys. 73(1931)376
38. M. L. Sage, J. Chem. Phys, 87(1984)431
39．朱正和，王风，许宗荣，原子与分子物理学报，2(1989)1007
40. F. L. Pilar. Elementary quantum chemistry. New York. McGraw-Hill Inc.,1968
41．孙卫国，侯世林等．双原子分子体系的振动结构研究．原子核物理评论．2002，19(20)：9l-94
42. YI Han-wei. ZHANG Feng-dong. Computing the pure vibrational and vibrational- rotational energy eigenvalues of ~7Li_2 by difference method, 光学精密工程. 2002, 10(3)
43. Cooney J P. Convenient Numerical Technique for Solving the One-dimensional Schrdinger Equationg for Bound states. Am. J. Phys. 1981, 49:76-77
44. Wu Lianao. Finite Differential Approach to the Radial Schrdinger Eqution. Acta Scientiarum Naturalium Universitatis Jilinensis. 1994,(3):67-70
45．徐献喻等．Padè逼近理论．上海：上海科学技术出版社，1989
46．马芳等．用无波函数微扰论和Padè近似方法计算双原子分子振动能级．辽宁大学学报(自然科学版)，1999，26(3)：233-237
47．冯霞，程小健．应用微扰论计算同核双原子分子非谐振子振转相互作用能及波函数．安徽师范大学学报(自然科学版)．2000，23(3)：227-230
48. M. D. Feit , J. A. Fleck , and A. Steiger. Solution of the Schrdinger Equationg by aspectral method. J. Comp. Phys. 1982,47:412
49. Richard Holland. Finite-Difference Solution of Maxwell's Equations In Generalized Nonorthogonal Coordinates. IEEE Trans. Nuclear Science. 1983, 30(6):4589-4595
50．李永明，俞集辉，黄键．时域有限差分法中的吸收边界条件与角点处理，重庆大学学报(自然科学版)．2001，24(3)61-71
51. MUR.G. Absorbing boundary Conditions for the finite Difference Approximation of the Time Domain Electromagnetic Field Equations. IEEE
Trans Microwave Theory Tech, 1981,MTT- 29(10):1073 1077.
52. ENG QUISTB. Absorbing boundary Conditions for the Numerical Simulation of wave. Mathematics of the Computation. 1977,31(139):629 651
53．王长清，祝西里．电磁场计算中的时域有限差分法．北京大学出版社．1994，IEEE．1997,85(7)：1031-1060．
54. Supriyo D , Mittra R. Efficient computation of resonant frequencies and quality factors of cavities via combination of the finite difference time domain technique and the Pade approximation. IEEE Microwave and Guided Wave Letters. 1998,8(12):415-417
55. Yee K , I ngham D , Shlager K. Time domain extra polation to the farfield based on FDTD calculations. IEEE Trans AP. 1991 , 39(4):429-433.
56. Gonzalez S, Garcia B, Garcia O lmedo R, etal. A time domain near to far field transformation for FDTD in two dimensions. Microwave and Optical Technology Letters. 2000,27(6):427-432
57. Cangellaris A. C, Zhao L. Rapid FDTD analysis of shielding enclosures. In: IEEE International Symposiumon Electromagnetic Compatibility. Seattle, WA, USA. 1999, 827-831.
58. Liao Cheng, Deng Y, RenL. Numerical solution for EM scattering about arbitrary two-dimensional bodies using Thom pson-FDTD method. Microwave & Opt. Tech. Lett. 1996, 13(4):233-236
59. LIAO Cheng, REN L. Comparison of Thom pson-FDTD method and canonical FDTD method. Chinese Journal of Radio Science. 1997, 12(4):356-360
60. Jia-Ying Wang. Ben-Qing Gao. A Method of Sub-Region Connection in FDTD Algorithm. Microwave And Optical Technology Letters. 1998, 9(3):229～233
61．朱正和，沈守瑶，牟炜明，李麟．分子科学与化学研究．1984，(3)：359
62. 62 K. P. Huber, G. Herzberg. Molecular spectra and molecular structure, Vol. IV , Constants of diatomic molecues . New York: Van Nostrand Reinhold Company. 1979.
63．李新喜．异核双原子分子势能函数的变分研究．四川大学硕士学位论文．2003．
2．许长存等，原子和分子光谱学，大连理工大学出版社，1989，164
3．曾谨言．量子力学．卷Ⅰ，第二版，北京：科学出版社，1997
4．喀兴林．高等量子力学．北京：高等教育出版社，1999
5. E. Schrdinger, Ann. Physik, 79 (1726), 361, 489; 80 (1926), 437; 81 (1926), 109.
6．朱正和，俞华根．分子结构与分子势能函数．北京：科学出版社，1997．
7. M.Born and J.R.Oppenheimer ,Ann. Physik, 1927, 84: 457.
8. M.Born and K.Huang. Dynamical Theory of Crystal Lattices. Oxford University Press. New York. 1954
9．唐敖庆，量子化学．科学出版社．1982．
10. N. V. Cohan , H. F. Hameka. Born Oppenheimer approximation and the calculation of infrared intensities. J. Chem. Phys. 1966,45:4392
11. M. Mills. Born Oppenheimer failure in sparation of low frequency molecular vibrations. J. Phys. Chem. 1984,88:532
12．徐办庄．分子光谱理论．北京：清化大学出版社，1998．169～174．
13．王国文．原子与分子光谱导论．北京：高等教育出版社，1985．
14. Yee K. S. Numerical solution of initial boundary value problems involving Maxwell' sequations in isotropic media. IEEE Trans. On A P. 1966 , 14:302-307
15．高本庆．《时域有限差分法FDTD Method》．国防工业出版社．1995
16. K. S. Kunz and K. M. Lee. A Three-Dimensional Finite-Differerce Solution of the External Response of an Aircraft to a Complex Transient EM Enviroment:Part 1-The Method and its Implementa Implementation. IEEE Trans. On Electromagnetic compitiblity, EMX-20. 1978, 328-332
17. A. Talove and M. E. Brodwin. Numerical solution of Steadystate Electromagmecic scattering Problems Vsing the Time-dependent Maxwell's Equations. IEEE Trans. Microware Theory Tech. MTT-23. 1975,623-630
18. A. Toflore. Application of the Finite-Difference Time-Domain Method to
Sinu soidal steady-state Electeomagnetic-Pentration Problems. IEEE Trans. On Electromagnetic Compotibility. 1980, EMC-22.191-202
19. K. Umashankar and A. Taflove, A Novel Method to analyzd Electromagnetic scattering of Complex objects. IEEE Trans. On Electromagnetic compitiblity, EMX-24.1981, 397-405
20. Choi D H and Hoefer W J R 1986 IEEE Trans. MTT-34 1464
21．黄整等．时域有限差分法对双原子分子振动光谱的计算的应用．原子分子物理学报．2000，17(4)：577．
22．罗有华等．冷原子在静电势阱中的量子力学效应。物理学报。2002，51(8)：1706-1709
23．凤尔银，郑贤锋．原子与分子物理学报．分立位置表象中双原子分子振动能级的计算．2002，19(3)：341-346
24．刘启能，冯灏．能量自洽法的建立以及对异核双原子分子的应用．原子与分子物理学报．2000，17(2)：339-342
25．刘启能．双原子分子振转能级的计算．四川师范大学学报(自然科学版)．1999．22(6)：730-734
26. R. Kosloff, Time dependent quantum mechanical methods for molecular dynamics. J. Phys. Chem. 1988,92:2087
27. K. P. Huber and G. Herzberg Constants of Diatomic Molecules, Van Nostrand, Princdton ,(1979)
28. J. N. Murrell and K. S. Sorbie ,J. Chem. Soc. Faraday Trans. Ⅱ,70(1974) 1552.
29. Y. P. Varshini ,Rev. Mod. phys.29(1957)664.
30. D. Steele, E. R. Lippincott and J.T . Vandensilce , Rev. Mod. Phys.34(1962)239.
31. I.N. Levine, J. Chem. Phys, 1984, (87):431.
32. Huxley P, Murrell J N. Ground state diatomic potentials. J. Chem. Soc. Faraday Trans. 1983, (79):323
33. Sun. Weiguo, FengHao. An Energy consistent Method for Potential Energy Curves of Diatomic Molecules. J Phys. 1999, B 32:5109
34. P. M. Morse , The potential curves of diatomic molecules. Phys. Rev. 1929,34:57
35. H. M. Hullbert and J. O. Hirschfelder, J. Chem. Phys, 34(1962)239
36. W. N. Whitton and P. J. Kuntz, J. Chem. Phys ,64(1976)3624
37. R. Rydberg, Z. Phys. 73(1931)376
38. M. L. Sage, J. Chem. Phys, 87(1984)431
39．朱正和，王风，许宗荣，原子与分子物理学报，2(1989)1007
40. F. L. Pilar. Elementary quantum chemistry. New York. McGraw-Hill Inc.,1968
41．孙卫国，侯世林等．双原子分子体系的振动结构研究．原子核物理评论．2002，19(20)：9l-94
42. YI Han-wei. ZHANG Feng-dong. Computing the pure vibrational and vibrational- rotational energy eigenvalues of ~7Li_2 by difference method, 光学精密工程. 2002, 10(3)
43. Cooney J P. Convenient Numerical Technique for Solving the One-dimensional Schrdinger Equationg for Bound states. Am. J. Phys. 1981, 49:76-77
44. Wu Lianao. Finite Differential Approach to the Radial Schrdinger Eqution. Acta Scientiarum Naturalium Universitatis Jilinensis. 1994,(3):67-70
45．徐献喻等．Padè逼近理论．上海：上海科学技术出版社，1989
46．马芳等．用无波函数微扰论和Padè近似方法计算双原子分子振动能级．辽宁大学学报(自然科学版)，1999，26(3)：233-237
47．冯霞，程小健．应用微扰论计算同核双原子分子非谐振子振转相互作用能及波函数．安徽师范大学学报(自然科学版)．2000，23(3)：227-230
48. M. D. Feit , J. A. Fleck , and A. Steiger. Solution of the Schrdinger Equationg by aspectral method. J. Comp. Phys. 1982,47:412
49. Richard Holland. Finite-Difference Solution of Maxwell's Equations In Generalized Nonorthogonal Coordinates. IEEE Trans. Nuclear Science. 1983, 30(6):4589-4595
50．李永明，俞集辉，黄键．时域有限差分法中的吸收边界条件与角点处理，重庆大学学报(自然科学版)．2001，24(3)61-71
51. MUR.G. Absorbing boundary Conditions for the finite Difference Approximation of the Time Domain Electromagnetic Field Equations. IEEE
Trans Microwave Theory Tech, 1981,MTT- 29(10):1073 1077.
52. ENG QUISTB. Absorbing boundary Conditions for the Numerical Simulation of wave. Mathematics of the Computation. 1977,31(139):629 651
53．王长清，祝西里．电磁场计算中的时域有限差分法．北京大学出版社．1994，IEEE．1997,85(7)：1031-1060．
54. Supriyo D , Mittra R. Efficient computation of resonant frequencies and quality factors of cavities via combination of the finite difference time domain technique and the Pade approximation. IEEE Microwave and Guided Wave Letters. 1998,8(12):415-417
55. Yee K , I ngham D , Shlager K. Time domain extra polation to the farfield based on FDTD calculations. IEEE Trans AP. 1991 , 39(4):429-433.
56. Gonzalez S, Garcia B, Garcia O lmedo R, etal. A time domain near to far field transformation for FDTD in two dimensions. Microwave and Optical Technology Letters. 2000,27(6):427-432
57. Cangellaris A. C, Zhao L. Rapid FDTD analysis of shielding enclosures. In: IEEE International Symposiumon Electromagnetic Compatibility. Seattle, WA, USA. 1999, 827-831.
58. Liao Cheng, Deng Y, RenL. Numerical solution for EM scattering about arbitrary two-dimensional bodies using Thom pson-FDTD method. Microwave & Opt. Tech. Lett. 1996, 13(4):233-236
59. LIAO Cheng, REN L. Comparison of Thom pson-FDTD method and canonical FDTD method. Chinese Journal of Radio Science. 1997, 12(4):356-360
60. Jia-Ying Wang. Ben-Qing Gao. A Method of Sub-Region Connection in FDTD Algorithm. Microwave And Optical Technology Letters. 1998, 9(3):229～233
61．朱正和，沈守瑶，牟炜明，李麟．分子科学与化学研究．1984，(3)：359
62. 62 K. P. Huber, G. Herzberg. Molecular spectra and molecular structure, Vol. IV , Constants of diatomic molecues . New York: Van Nostrand Reinhold Company. 1979.
63．李新喜．异核双原子分子势能函数的变分研究．四川大学硕士学位论文．2003．