臺灣博碩士論文加值系統

在這份論文中,我們利用線性半古典初始值表示法來研究一些凝態中的動態過程及吸收光譜。首先,我們運用這個方法來處理一個弗蘭克爾激子模型,藉著跟數值上精確的準絕熱路徑積分法比較,我們從系統在某個態隨著時間變化的期望值可以了解這個方法的適用範圍。我們的計算結果指出這個方法在所有考慮的參數範圍都給予了正確的 振盪頻率但錯誤的平衡位置,此外,我們觀察到當系統的激子態能階差大於環境普密度的截止頻率時,去相干速率可以被精確的描述。這個方法所有的偏差都可以它採用了半古典假設去解釋。我們更進一步的針對平衡位置的偏差提出修正的方法,並且在大部份的參數範圍都得到了非常準確的結果。在第二部分,我們延伸了這個方法到吸收光譜的計算。我們證實了它在單體以及二聚體系統給予了精確的結果。另外,我們更針對了現實生活的複雜系統計算出合理的結果。最後一部份,我們利用這個方法來處理一個朗之萬耗散的系統,藉此我們觀察到馬庫斯定律的適用範圍在時域上確實有個上界以及下界,我們更系統化的去分析對於不同電子予體態以及電子受體態的能階差,這個範圍的變化,此外我們還找出了對於所有能接差都通用的範圍。

In this thesis, we perform theoretical investigations on dynamical processes in condensed phases and spectra by utilizing the linearized semiclassical initial value representation (LSC-IVR) in the Meyer-Miller representation. First, we apply this method to the Frenkel exciton model, determining its applicable regime by comparing the population dynamics with numerically exact quasi adiabatic path integral results. Our calculations suggest that this method gives correct oscillating frequencies but incorrect equilibrium populations. Besides, the decoherence rate are well described as long as the excitonic energy gap is larger compared with the cut-off frequency. We conclude that all the deviations from exact results can be elucidated from the classical approximation in the LSC-IVR method. Moreover, we provide a long time correction, successfully modifying the equilibrium position. Second, we dis- cuss the validity of the LSC-IVR method in simulating the absorption spectra. We demonstrate that it gives excellent results in both monomer and dimer systems. By utilizing the property of less computational expense in this approach, we also successfully reproduce the reasonable absorption lineshape for real complex system. Finally, we consider a dissipative Hamiltonian with a random force and friction yielding a form of Langevin dynamics, where the propagation is based on the LSC-IVR approach, to study the emergence of the Marcus theory. Our results show that there exists an lower and upper bounds in time domain for obtaining Marcus rates. We further find out an universal regime which is suitable for different driving forces.

口試委員會審定書
致謝
中文摘要
Abstract
Contents
List of Figures
1 Introduction 1
2 Theoretical Background 4
2.1 SC-IVR................................... 4
2.2 LSC-IVR .................................. 6
2.3 Meyer-Miller-Stock-ThossRepresentation . . . . . . . . . . . . . . . . . 9
3 LSC-IVR Simulation of the Frenkel Exciton Model 12
3.1 FrenkelExcitonModel ........................... 12
3.2 BathDiscretization ............................. 14
3.3 Pure-DephasingModel ........................... 16
3.4 Conclusion ................................. 21
4 Benchmark the Performance of the LSC-IVR 22
4.1 TheoreticalModel.............................. 22
4.2 ComparisonwithQUAPI.......................... 23
4.3 LongTimeCorrection ........................... 27
4.4 Conclusion ................................. 31
5 Absorption Spectra 33
5.1 TheoreticalModel.............................. 33
5.2 Monomer Linearly Coupled to a Single Vibrational Mode . . . . . . . . . 35
5.3 DimerLinearlyCoupledtoN HarmonicModes . . . . . . . . . . . . . . 37
5.4 FMO..................................... 39
5.5 Conclusion ................................. 41
6 The Emergence of the Marcus Theory 43
6.1 Introduction................................. 43
6.2 TheoreticalModel.............................. 45
6.3 ResultsandDiscussion ........................... 46
6.4 Conclusion ................................. 52
A The Derivation of the van Vleck Propagator 54
Bibliography 58

[1] X. Daura, B. Jaun, D. Seebach, W. F. van Gunsteren, and A. E. Mark, “Re- versible peptide folding in solution by molecular dynamics simulation,” J. Mol. Biol., vol. 280, p. 925, 1998.
[2] M.KarplusandG.A.Petsko,“Moleculardynamicssimulationsinbiology,”Nature, vol. 347, p. 631, 1990.
[3] S. B. Legoas, V. R. Coluci, S. F. Braga, P. Z. Coura, S. O. Dantas, and D. S. Gal- vao, “Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators,” Phys. Rev. Lett., vol. 90, p. 055504, 2003.
[4] J. L. Klepeis, K. Lindorff-Larsen, R. O. Dror, and D. E. Shaw, “Long-timescale molecular dynamics simulations of protein structure and function,” Curr. Opin. Struct. Biol, vol. 19, p. 120, 2009.
[5] J.C.Tully,“Moleculardynamicswithelectronictransitions,”J.Chem.Phys.,vol.93, p. 1061, 1990.
[6] J. C. Tully, “Mixed quantum–classical dynamics,” Faraday Discuss., vol. 110, p. 407, 1998.
[7] R. Kapral and G. Ciccotti, “Mixed quantum-classical dynamics,” J. Chem. Phys., vol. 110, p. 8919, 1999.
[8] R. Kapral, “Progress in the theory of mixed quantum-classical dynamics,” Annu. Rev. Phys. Chem., vol. 57, p. 129, 2006.
[9] G.STOCKandM.THOSS,“Classicaldescriptionofnonadiabaticquantumdynam- ics,” Adv. Chem. Phys., vol. 131, p. 243, 2005.
[10] X. Sun, H. Wang, and W. H. Miller, “Semiclassical theory of electronically nonadi- abatic dynamics: Results of a linearized approximation to the initial value represen- tation,” J. Chem. Phys., vol. 109, p. 7064, 1998.
[11] W.H.Miller,“TheSemiclassicalInitialValueRepresentation: APotentiallyPracti- cal Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations,” J. Phys. Chem. A, vol. 105, p. 2942, 2001.
[12] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Sta- tistical Mechanics, 2nd Edition. Springer, 1998.
[13] B. J. BERNE and G. D. HARP, “ON THE CALCULATION OF TIME CORRELA- TION FUNCTIONS,” Adv. Chem. Phys., vol. XVII, p. 63, 1970.
[14] N. Makri and K. Thompson, “Semiclassical influence functionals for quantum sys- tems in anharmonic environments,” Chem. Phys. Lett., vol. 291, p. 101, 1998.
[15] K. Thompson and N. Makri, “Influence functionals with semiclassical propagators in combined forward–backward time,” J. Chem. Phys., vol. 110, p. 1343, 1999.
[16] H. Wang, X. Sun, and W. H. Miller, “Semiclassical approximations for the calcula- tion of thermal rate constants for chemical reactions in complex molecular systems,” J. Chem. Phys., vol. 108, p. 9726, 1998.
[17] S.Mukamel,PrinciplesofNonlinearOpticalSpectroscopy.OxfordUniversityPress, 1999.
[18] R.E.FennaandB.W.Matthews,“Chlorophyllarrangementinabacteriochlorophyll protein from Chlorobium limicola,” Nature, vol. 258, p. 573, 1975.
[19] R. Marcus, “On the theory of oxidation reduction reactions involving electron trans- fer. i,” J. Chem. Phys., vol. 24, p. 966, 1956.
[20] R. Marcus, “Theoretical relations among rate constants, barriers, and theoretical re- lations among rate constants, barriers, and bronsted slopes of chemical reactions,” J. Phys. Chem., vol. 72, p. 891, 1968.
[21] R. A. Marcus and N. Sutin, “Electron transfers in chemistry and biology,” Biochimi. Biophys. Acta, vol. 811, p. 265, 1985.
[22] A. J. Leggett, “Quantum tunneling in the presence of an arbitrary linear dissipation mechanism,” Phys. Rev. B, vol. 30, p. 1208, 1984.
[23] A. Garg, J. N. Onuchic, and V. Ambegaokar, “Effect of friction on electron transfer in biomolecules,” J. Chem. Phys., vol. 83, p. 4491, 1985.
[24] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals. McGraw- Hill Companies, 1965.
[25] N. Makri, “Feynman path integration in quantum dynamics,” Comput. Phys. Com- mun., vol. 63, p. 389, 1991.
[26] J.H.vanVleck,“THECORRESPONDENCEPRINCIPLEINTHESTATISTICAL INTERPRETATION OF QUANTUM MECHANICS,” PNAS, vol. 14, p. 178, 1928.
[27] H.-D.MeyerandW.H.Miller,“Aclassicalanalogforelectronicdegreesoffreedom in nonadiabatic collision processes,” J. Chem. Phys., vol. 70, p. 3214, 1979.
[28] G. Stock and M. Thoss, “Semiclassical Description of Nonadiabatic Quantum Dy- namics,” Phys. Rev. Lett., vol. 78, p. 578, 1997.
[29] M.ThossandG.Stock,“Mappingapproachtothesemiclassicaldescriptionofnona- diabatic quantum dynamics,” Phys. Rev. A, vol. 59, p. 64, 1999.
[30] V. May and O. Ku&;#776;hn, Charge and Energy Transfer Dynamics in Molecular Systems, 3rd, Revised and Enlarged Edition. Wiley-VCH, 2011.
[31] J. A. Leegwater, “Coherent versus Incoherent Energy Transfer and Trapping in Pho- tosynthetic Antenna,” J. Phys. Chem., vol. 100, p. 14403, 1996.
[32] G.Panitchayangkoon,D.Hayes,K.A.Fransted,J.R.Caram,E.Harel,J.Wen,R.E. Blankenship, and G. S. Engel, “Long-lived quantum coherence in photosynthetic complexes at physiological temperature,” PNAS, vol. 107, p. 12766, 2010.
[33] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Manc&;#780;al, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature, vol. 446, p. 782, 2007.
[34] A.J.Leggett,S.Chakravarty,A.T.Dorsey,M.P.A.Fisher,A.Garg,andW.Zwerger, “Dynamics of the dissipative two-state system,” vol. 59, p. 1, 1987.
[35] H. Wang, X. Song, D. Chandler, and W. H. Miller, “Semiclassical study of elec- tronically nonadiabatic dynamics in the condensed-phase: Spin-boson problem with Debye spectral density,” J. Chem. Phys., vol. 110, p. 4828, 1999.
[36] J.L.SkinnerandD.Hsu,“PureDephasingofaTwo-LevelSystem,”J.Phys.Chem., vol. 90, p. 4931, 1986.
[37] G. Tao and W. H. Miller, “Semiclassical Description of Electronic Excitation Pop- ulation Transfer in a Model Photosynthetic System,” J. Phys. Chem. Lett., vol. 1, no. 6, p. 891, 2010.
[38] U. Weiss, Quantum Dissipative Systems, 4th Edition. World Scientific Publishing Company, 2012.
[39] N. Makri and D. E. Makarov, “Tensor propagator for iterative quantum time evo- lution of reduced density matrices. I. Theory,” J. Chem. Phys., vol. 102, p. 4600, 1995.
[40] N. Makri and D. E. Makarov, “Tensor propagator for iterative quantum time evo- lution of reduced density matrices. II. Numerical methodology,” J. Chem. Phys., vol. 102, p. 4611, 1995.
[41] H.-T. Chang, P.-P. Zhang, and Y.-C. Cheng, “Criteria for the accuracy of small po- laron quantum master equation in simulating excitation energy transfer dynamics,” J. Chem. Phys., vol. 139, p. 224112, 2013.
[42] L.Chen,R.Zheng,Q.Shi,andY.Yan,“Opticallineshapesofmolecularaggregates: Hierarchical equations of motion method,” J. Chem. Phys., vol. 131, p. 094502, 2009.
[43] Y. Tanimura and R. Kubo, “Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath,” J. Phys. Soc. Jpn., vol. 58, p. 101, 1989.
[44] A. Ishizaki and G. R. Fleming, “Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature,” PNAS, vol. 106, p. 17255, 2009.
[45] S. I. E. Vulto, M. A. de Baat, R. J. W. Louwe, H. P. Permentier, T. Neef, M. Miller, H. van Amerongen, and T. J. Aartsma, “Exciton Simulations of Optical Spectra of the FMO Complex from the Green Sulfur Bacterium Chlorobium tepidumat 6 K,” J. Phys. Chem. B, vol. 102, p. 9577, 1998.
[46] E. L. Read, G. S. Schlau-Cohen, G. S. Engel, J. Wen, R. E. Blankenship, and G. R. Fleming, “Visualization of Excitonic Structure in the Fenna-Matthews-Olson Pho- tosynthetic Complex by Polarization-Dependent Two-Dimensional Electronic Spec- troscopy,” Biophys. J., vol. 95, p. 847, 2008.
[47] B. R. Landry and J. E. Subotnik, “Communication: Standard surface hopping pre- dicts incorrect scaling for Marcus’ golden-rule rate: The decoherence problem can- not be ignored,” J. Chem. Phys., vol. 135, p. 191101, 2011.
[48] W. Xie, S. Bai, L. Zhu, and Q. Shi, “Calculation of Electron Transfer Rates Using Mixed Quantum Classical Approaches: Nonadiabatic Limit and Beyond,” J. Phys. Chem. A, vol. 117, p. 6196, 2013.
[49] I. Prigogine and S. A. Rice, “Electron transfer reactions in solution: A historical per- spective,” Adv. Chem. Phys.: Electron Transfer - from Isolated Molecules to Biom- elecules, Part 1 and 2, vol. 106, 2007.
[50] H. B. Gray and J. R. Winkler, “ELECTRON TRANSFER IN PROTEINS,” Annu. Rev. Biochem., vol. 65, p. 537, 1996.
[51] P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Proc. R. Soc. A, vol. 114, p. 243, 1927.
[52] H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechan- ics. Springer Tracts in Modern Physics, 1982.
[53] D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, 2006.
[54] S. Levit and U. Smilansky, “A theorem on infinite products of eigenvalues of sturm- liouville type operators,” vol. 65, p. 299, 1977.
[55] M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics. Springer, 1991.