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作者(中文):王 瑜
作者(外文):Wang, Yu
論文名稱(中文):以近似玻姆軌跡法模擬非絕熱動力學
論文名稱(外文):Simulating nonadiabatic dynamics with approximate Bohmian trajectories
指導教授(中文):周佳駿
指導教授(外文):Chou, Chia-Chun
口試委員(中文):朱立岡
鄭博元
口試委員(外文):Chu, Li-Kang
Cheng, Po-Yuan
學位類別:碩士
校院名稱:國立清華大學
系所名稱:化學系
學號:104023544
出版年(民國):106
畢業學年度:105
語文別:英文
論文頁數:68
中文關鍵詞:非絕熱動力學玻姆軌跡法
外文關鍵詞:nonadiabaticBohmiantrajectories
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非絕熱動力學是許多分子的物理及化學過程的基礎。然而,這樣的系統需要經過一定程度的近似,才能被有效的模擬。我們改善了量子軌跡法(QTM),使其能在維持一定的效能以及精準度的情形下,來處理非絕熱系統。我們使用了複數量子哈密頓-雅可比方程式搭配玻姆軌跡(CQHJE-BT),以及導數傳遞法(DPM)來處理非絕熱系統。我們在一維以及二維的系統都得到良好的結果。波函數的振幅以及相位都被精確的模擬。儘管仍然有許多問題待處理,我們仍然一定程度的改善了量子軌跡法,使其能有效且精確的處理更複雜的系統。
Nonadiabatic dynamics is the core of various molecular processes. However, approximations are needed for efficiency considerations. We improved the quantum trajectory method (QTM) significantly while retaining its accuracy on nonadiabatic systems. The complex quantum Hamilton-Jacobi equations with Bohmian trajectories (CQHJE-BT) and the derivative propagation method (DPM) were applied to two-state nonadiabatic systems. Results showed that this method are capable of dealing with not only one-dimensional but also two-dimensional systems. Both the amplitude and the phase of the wave function can be evaluated accurately by CQHJE-BT. However, there are still some aspects needed to be improved. Still, the QTM has been improved to deal with more complex systems while retaining efficiency and accuracy.
1 Introduction 5
2 Representations in nonadiabatic dynamics 7
2.1 Factoring the molecular state . . . . . . . . . . . . . . . . . . . . . 7
2.2 Diabatic representation . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Born-Oppenheimer adiabatic representation . . . . . . . . . . . . . 11
2.4 Adiabatic approximations . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Transformation between representations . . . . . . . . . . . . . . . 14
2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Quantum Hydrodynamics 19
3.1 The Madelung-Bohm approach . . . . . . . . . . . . . . . . . . . 19
3.2 Hydrodynamic view of quantum dynamics . . . . . . . . . . . . . 20
3.3 Trajectories in quantum hydrodynamics . . . . . . . . . . . . . . . 23
3.4 The complex quantum Hamilton-Jacobi equation . . . . . . . . . . 24
3.5 Derivative propagation method . . . . . . . . . . . . . . . . . . . 26
4 Model problems and computational results 31
4.1 Model systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Results of the CQHJE-BT method . . . . . . . . . . . . . . . . . 38
4.3.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Discussion and conclusion 49
5.1 Failure of higher order DPM . . . . . . . . . . . . . . . . . . . . . 49
5.2 Node problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
A Change of Basis 55
B Derivative propagation equations 59
Bibliography 63
[1] M. Baer, Beyond Born–Oppenheimer : electronic nonadiabatic coupling terms and
conical intersections, (John Wiley & Sons, New Jersy, 2006).
[2] L. S. Cederbaum, “Born–Oppenheimer approximation and beyond for timedependent
electronic processes,” J. Chem. Phys. 128, 124101 (2008).
[3] J. C. Tully, “Molecular dynamics with electronic transitions,” J. Chem. Phys.
93, 1061 (1990).
[4] A. Donoso and C. C. Martens, “Semiclassical multistate Liouville dynamics in
the adiabatic representation,” J. Chem. Phys. 112, 3980 (2000).
[5] A. Abedi, N. T. Maitra, and E. K. U. Gross, “Exact Factorization of the Time-
Dependent Electron-Nuclear Wave Function,” Phys. Rev. Lett. 105, 123002
(2010).
[6] C.-Y. Zhu, “Restoring electronic coherence/decoherence for a trajectory-based
nonadiabatic molecular dynamics,” Sci. Rep. 6, 24198 (2016).
[7] C. L. Lopreore and R. E. Wyatt, “Quantum Wave Packet Dynamics with Trajectories,”
Phys. Rev. Lett. 82, 5190 (1999).
[8] C.-C. Chou, “Complex quantum Hamilton-Jacobi equation with Bohmian trajectories:
Application to the photodissociation dynamics of NOCl,” J. Chem.
Phys. 140, 104307 (2014).
[9] C. L. Lopreore and R. E. Wyatt, “Electronic transitions with quantum trajectories.
II,” J. Chem. Phys. 116, 1228 (2002).
[10] R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics,
(Springer, New York, 2005).
[11] Y. Goldfarb, I. Degani, and D. J. Tannor, “Bohmian mechanics with complex
action: A new trajectory-based formulation of quantum mechanics,” J. Chem.
Phys. 125, 231103 (2006).
[12] R. A. Leacock and M. J. Padgett, “Hamilton-Jacobi Theory and the Quantum
Action Variable,” Phys. Rev. Lett. 50, 3 (1983).
[13] R. A. Leacock and M. J. Padgett, “Hamilton-Jacobi/action-angle quantum mechanics,”
Phys. Rev. D 28, 2491 (1983).
[14] C. J. Trahan, K. Hughes, and R. E. Wyatt, “A new method for wave packet
dynamics: Derivative propagation along quantum trajectories,” J. Chem. Phys.
118, 9911 (2003).
[15] T. Azumi and K. Matsuzaki, “What Does the Term ”Vibronic Coupling”
Mean?” Photochem. Photobiol. 25, 315 (1977).
[16] F. T. Smith, “Diabatic and Adiabatic Representations for Atomic Collision Problems,”
Phys. Rev. 179, 111 (1969).
[17] M. Baer, “Electronic non-adiabatic transitions: Derivation of the general
adiabatic-diabatic transformation matrix,” Mol. Phys. 40, 1011 (1980).
[18] B. T. Sutcliffe and R. G. Woolley, “Molecular structure calculations without
clamping the nuclei,” Phys. Chem. Chem. Phys. 7, 3664 (2005).
[19] Á. S. Sanz and S. Miret-Artés, A Trajectory Description of Quantum Processes. I.
Fundamentals, (Springer, Berlin, 2012).
[20] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics,
volume 3, (Addison-Wesley, San Francisco, 2013).
[21] E. R. Bittner and R. E. Wyatt, “Integrating the quantum Hamilton–Jacobi
equations by wavefront expansion and phase space analysis,” J. Chem. Phys.
113, 8888 (2000).
[22] N. Zamstein and D. J. Tannor, “Non-adiabatic molecular dynamics with complex
quantum trajectories. I. The diabatic representation,” J. Chem. Phys. 137,
22A517 (2012).
[23] R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, (Brooks/Cole,
Boston, 2011).
[24] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary
Value Problems, 10th edition, (John Wiely & Sons, New York, 2013).
[25] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd edition,
(Addison-Wesley, San Francisco, 2011).
[26] D. J. Tannor, Introduction to Quantum Mechanics: a Time-Dependent Perspective,
(University Science Books, Sausalito, 2007).
[27] P. K. Kundu, I. M. Cohen, and D. R. Dowling, Fluid Mechanics, 6th edition,
(Elsevier, San Diego, 2016).
[28] H. Goldstein, C. P. P. Jr., and J. L. Safko, Classical Mechanics, 3rd edition,
(Pearson, Addison-Wesley, 2001).
[29] R. E. Wyatt, C. L. Lopreore, and G. Parlant, “Electronic transitions with quantum
trajectories,” J. Chem. Phys. 114, 5113 (2001).
[30] N. Zamstein and D. J. Tannor, “Non-adiabatic molecular dynamics with complex
quantum trajectories. II. The adiabatic representation,” J. Chem. Phys. 137,
22A518 (2012).
[31] C.-C. Chou, “Complex quantum Hamilton–Jacobi equation with Bohmian
trajectories for wave packet dynamics,” Chem. Phys. Lett. 591, 203 (2014).
[32] V. A. Rassolov and S. Garashchuk, “Semiclassical nonadiabatic dynamics with
quantum trajectories,” Phys. Rev. A 71, 032511 (2005).
[33] R. E. Wyatt and K. Na, “Quantum trajectory analysis of multimode subsystembath
dynamics,” Phys. Rev. E 65, 016702 (2001).
[34] F. S. Mayor, A. Askar, and H. A. Rabitz, “Quantum fluid dynamics in the
Lagrangian representation and applications to photodissociation problems,” J.
Chem. Phys. 111, 2423 (1999).
[35] R. E. Wyatt and B. A. Rowland, “Computational Investigation of Wave Packet
Scattering in the Complex Plane: Propagation on a Grid,” J. Chem. Theory
Comput. 5, 443 (2009).
[36] D. Kohena, F. H. Stillinger, and J. C. Tully, “Model studies of nonadiabatic
dynamics,” J. Chem. Phys. 109, 4713 (1998).
[37] J. C. Burant and J. C. Tully, “Nonadiabatic dynamics via the classical limit
Schrödinger equation,” J. Chem. Phys. 112, 6097 (2000).
[38] C. A. Mead and D. G. Truhlar, “On the determination of Born–Oppenheimer
nuclear motion wave functions including complications due to conical intersections
and identical nuclei,” J. Chem. Phys. 70, 2284 (1979).
[39] C. A. Mead and D. G. Truhlar, “Conditions for the definition of a strictly diabatic
electronic basis for molecular systems,” J. Chem. Phys. 77, 6090 (1982).
[40] M. Baer, “Adiabatic and Diabatic Representations for Atom-Molecule Collisions:
Treatment of the Collinear Arrangement,” Chem. Phys. Lett. 35, 112
(1975).
[41] M. Baer, “Introduction to the theory of electronic non-adiabatic coupling terms
in molecular systems,” Phys. Rep. 358, 75 (2002).
[42] C. J. Ballhausen and A. E. Hansen, “Electronic Spectra,” Annu. Rev. Phys.
Chem. 23, 15 (1972).
[43] R. Kapral, “Progress in the Theory of Mixed Quantum-Classical Dynamics,”
Annu. Rev. Phys. Chem. 57, 129 (2006).
[44] T. V. Voorhis, T. Kowalczyk, B. Kaduk, L.-P. Wang, C.-L. Cheng, and Q. Wu,
“The Diabatic Picture of Electron Transfer, Reaction Barriers, and Molecular
Dynamics,” Annu. Rev. Phys. Chem. 61, 149 (2010).
[45] A. W. Jasper, C. Zhu, S. Nangia, and D. G. Truhlar, “Introductory lecture:
Nonadiabatic effects in chemical dynamics,” Faraday Discuss. 127, 1 (2004).
[46] M. Baer and R. Englman, “A study of the diabatic electronic representation
within the Born-Oppenheimer approximation,” Mol. Phys. 75, 293 (1992).
[47] J. P. Malhado, M. J. Bearpark, and J. T. Hynes, “Non-adiabatic Dynamics Close
to Conical Intersections and the Surface Hopping Perspective,” Front. Chem.
2, 1 (2014).
[48] M. Born and J. R. Oppenheimer, “Zur Quantentheorie der Molekeln,” Ann.
Phys. 389, 457 (1927).
[49] B. T. Sutcliffe and R. G. Woolley, “On the Quantum Theory of Molecules,” J.
Chem. Phys. 137, 22A544 (2012).
[50] B. T. Sutcliffe and R. G. Woolley, “Comment on ”On the quantum theory of
molecules”,” J. Chem. Phys. 140, 037101 (2014).
[51] T. Jecko, “On the mathematical treatment of the Born-Oppenheimer approximation,”
J. Math. Phys. 55, 053504 (2014).
[52] C. Wittig, “The Landau-Zener Formula,” J. Phys. Chem. B 109, 8428 (2005).
[53] J. R. Rubbmark, M. M. Kash, M. G. Littman, and D. Kleppner, “Dynamical
effects at avoided level crossings: A study of the Landau-Zener effect using
Rydberg atoms,” Phys. Rev. A 23, 3107. (1981).
[54] M. Wilkinson and M. A. Morgan, “Nonadiabatic transitions in multilevel systems,”
Phys. Rev. A 61, 062104 (2000).
[55] C. P. Sun, X. F. Liu, D. L. Zhou, and S. X. Yu, “Quantum measurement via
Born-Oppenheimer adiabatic dynamics,” Phys. Rev A 63, 012111 (2000).
[56] S. H. Lin and H. Eyring, “Study of Vibronic and Born-Oppenheimer Couplings,”
Proc. Natl. Acad. Sci. U.S.A. 71, 3415 (1974).
[57] E. Deumens, A. Diz, R. Longo, and Y. Öhrn, “Time-dependent theoretical
treatments of the dynamics of electrons and nuclei in molecular systems,” Rev.
Mod. Phys. 66, 917 (1994).
[58] M. V. Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,” Proc.
R. Soc. Lond. A Math. Phys. Sci. 392, 45 (1984).
[59] A. Mostafazadeh, “Quantum adiabatic approximation and the geometric phase,”
Phys. Rev. A 5, 1653 (1997).
[60] V. A. Rassolov and S. Garashchuk, “Quantum adiabatic approximation, quantum
action, and Berry’s phase,” Phys. Lett. A 232, 395 (1997).
[61] R. Resta, “Manifestations of Berry’s phase in molecules and condensed matter,”
J. Phys. Condens. Matter 12, R107 (2000).
[62] C. A. Mead, “The geometric phase in molecular systems,” Rev. Mod. Phys. 64,
51 (1992).
[63] A. Donoso, D. Kohen, and C. C. Martens, “Simulation of nonadiabatic wave
packet interferometry using classical trajectories,” J. Chem. Phys. 112, 7345
(2000).
[64] A. Donoso and C. C. Martens, “Classical Trajectory-Based Approaches to Solving
the Quantum Liouville Equation,” Int. J. Quantum Chem. 90, 1348 (2002).
[65] A. Donoso and C. C. Martens, “Simulation of Coherent Nonadiabatic Dynamics
Using Classical Trajectories,” J. Phys. Chem. A 102, 4291 (1998).
[66] A. Abedi, N. T. Maitra, and E. K. U. Gross, “Correlated electron-nuclear dynamics:
Exact factorization of the molecular wavefunction,” J. Chem. Phys.
137, 22A530 (2012).
[67] F. Agostini, A. Abedi, and E. K. U. Gross, “Classical nuclear motion coupled to
electronic non-adiabatic transitions,” J. Chem. Phys. 141, 214101 (2014).
[68] F. Agostini, A. Abedi, Y. Suzuki, S. K. Min, N. T. Maitra, and E. K. U. Gross,
“The exact forces on classical nuclei in non-adiabatic charge transfer,” J. Chem.
Phys. 142, 084303 (2015).
[69] B. F. E. Curchod, F. Agostini, and E. K. U. Gross, “An exact factorization perspective
on quantum interferences in nonadiabatic dynamics,” J. Chem. Phys.
145, 034103 (2016).
[70] F. Agostini, S. K. Min, A. Abedi, and E. K. U. Gross, “Quantum-Classical Nonadiabatic
Dynamics: Coupled- vs Independent-Trajectory Methods,” J. Chem.
Theory Comput. 12, 2127 (2016).
[71] A. Schild, F. Agostini, and E. K. U. Gross, “Electronic Flux Density beyond the
Born−Oppenheimer Approximation,” J. Phys. Chem. A 120, 3316 (2016).
[72] Y. Suzuki, A. Abedi, N. T. Maitra, K. Yamashita, and E. K. U. Gross, “Electronic
Schr¨odinger equation with nonclassical nuclei,” Phys. Rev. A 89, 040501
(2014).
[73] R. Requist, F. Tandetzky, and E. K. U. Gross, “Molecular geometric phase from
the exact electron-nuclear factorization,” Phys. Rev. A 93, 042108 (2016).
[74] S. K. Min, A. Abedi, K. S. Kim, and E. K. U. Gross, “Is the Molecular Berry
Phase an Artifact of the Born-Oppenheimer Approximation?” Phys. Rev. Lett.
113, 123002 (2014).
[75] S. K. Min, F. Agostini, and E. K. U. Gross, “Coupled-Trajectory Quantum-
Classical Approach to Electronic Decoherence in Nonadiabatic Processes,” Phys.
Rev. Lett. 115, 073001 (2015).
[76] P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA recommended values
of the fundamental physical constants: 2014,” Rev. Mod. Phys. 88, 035009
(2016).
 
 
 
 
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