臺灣博碩士論文加值系統

化學的核心領域之一是研究化學反應，理論計算提供實驗學家另一個觀點去詮釋它。在一般的化學反應裡，因為過渡態（過渡態可以評估化學反應的機制）的生命期很短，實驗上並不容易測得。然而，在理論計算與超級電腦的幫助下，過渡態可以被預估，進而提供預測反應機制的可能性。本篇論文旨在探討一種特殊的化學反應：分岐反應，它的選擇性不能被經典的過渡態理論解釋。另外兩個短篇的研究分別為質子偶合電子的轉移反應以及震動運動中的簡諧性的回顧。

分歧反應是主要研究計畫，在本篇論文的第一章至第六章，以及附錄Ａ、Ｂ和Ｃ。一個過渡態卻產生兩個產物是為分歧反應。有機化學家對自然界如何選擇分歧反應的產物的機制產生極大的興趣（第一章）。從理論化學家的觀點，這個反應需要多維度的位能曲面（第二章）。傳統的計算方法利用分子動力學模擬多維度的反應（第三章），但是如何分析分子動力學產生的軌跡需要反應位能面的幫助。我們在第四章定義一個人工的反應座標，來建立非對稱性的分歧反應的二維位能曲面。在第五章，使用此位能曲面分析分子動力學的軌跡。這個分析方法需要將位能曲面上的分子結構，與分子動態學的軌跡的分子結構，擺放在相同的坐標系下，因此在附錄A，我們提供其公式推導以及程式碼連結。最後，第六章應用上述方法模擬有機分子的分歧反應的二維位能曲面及軌跡，此曲面的形狀吻合軌跡的模擬結果，這些頃斜的位能曲面可以詮釋非對稱性的分歧反應的選擇性。

第七章使用質子和電子偶合的經典系統模擬其結果。質子和電子偶合的轉移反應定義為質子和電子同時從提供者轉移至接受者。因為電子和質子偶合，常用的波恩-歐本海默假設並不成立，因此必須使用因此必須使用非絕熱態替代絕熱態。我們使用直接偶和方法建構非絕熱態，而此方法不需要做絕熱態與非絕熱態的轉換。

第八章描述非簡諧振動的推導細節。在簡諧假設框架之外，非簡諧在分子震動貢獻相當重要的角色，例如化學鍵的伸長或解離。我們利用擾動理論分析非簡諧性的貢獻至第二擾動理論，其中，紛擾動的哈密爾敦函數使用簡諧假設，而擾動的算符使用不同簡正模的偶合建構。

附錄Ｃ描述轉動運動的推導及技術細節，因為校正兩個物體在空間中的位置與方向是分析軌跡的必要細節（第五章）。附錄Ｄ描述電子結構理論的概要背景，此為描述電子如何在分子中運動的基礎。因為電子是費米子，它們遵守一些量子力學中的公理，並且在任何化學鍵的斷裂或生成都扮演重要的角色。另一方面，視系統的特性來看，原子核可以使用古典力學處理（第一章）或是使用量子力學處理（第七章）。因此，附錄Ｅ簡介如何使用離散變數表徵處理質子的波函數。

Chemical reactions are the heart of chemistry, and they are no longer elusive
or magical in small flasks with the advent of theoretical and computational chemistry. For a given chemical reaction, the transition state is hard to observe for experimental chemists because of the extremely short life time which is beyond the limitation for most of traditional instruments. It is possible to reveal the structure and other properties of the transition state through current powerful super-computers and reasonable approximations from quantum chemistry. For exploring chemical reactions, we put an emphasis on the reactions with post-transition-state bifurcation (PTSBs), where its selectivity cannot be explained by conventional transition-state theory. Another two minor projects are also included; proton-coupled electron transfer (PCET) reactions and the study of anharmonicity for vibration.

PTSB is the main research project in this dissertation, and it is included in parts I to IV (chapters 1 to 6), and appendices A to C. Here, a single transition state bifurcates into two product states. How the selectivity for the product is determined has been an interesting problem for organic chemists (chapter 1). From the perspective of a theoretical chemist, this problem involves developing a multi-dimensional potential energy surface (chapter 2). Traditional methods employ molecular dynamics to capture all the degrees of freedom (chapter 3), but analyzing the resulting trajectories requires the existence of potential energy hypersurfaces. We propose a method to build two-dimensional potential energy surfaces (2D-PESs) for general PTSBs in chapter 4, and maps trajectories on the above PESs in chapter 5. The mapping between static PESs and dynamic trajectories requires the transformation of the coordinate of the structures belonging to the trajectories into the same Cartesian coordinate system as that of the PES, which is elucidated in appendix C. Finally, we illustrates the above approach with representative examples from the field of organic chemistry in chapter 6.

In chapter 7, a detailed discussion of PCET prototype systems was included. PCET involves transfer of both protons and electrons simultaneously from donor to acceptor. Since electrons and protons are coupled, adiabatic approximation is not suitable in this scenario and it is imperative to replace adiabatic states with diabatic states. Here, we use direct coupling (DC) to build diabatic states.

Beyond harmonic approximation, anharmonicity plays a crucial role in vibration especially in the bond stretching/dissociations process at high frequencies in the potential energy surface. This effect of polyatomic molecule is investigated using second-order perturbation theory, where unperturbed Hamiltonian uses harmonic approximation and perturbation operator is constructed by coupling with different normal modes. In chapter 8, we elaborates the detail derivation of above processes which is usually ignored in the literature.

In appendix C, we reviews the derivation and technical procedure of rotation, where alignment of two objects is necessary to map trajectories (chapter 5). In appendix D, we gives a brief background of electronic structure theory which is the foundation to describe how electrons move. Since electrons are fermions, they follow certain axioms in quantum mechanics and are involved in any bond formation/breaking process. On the other hand, nuclei can be treated classically (chapter 1) or quantum mechanically (chapter 7) on different limited conditions. In appendix E, we describes how to build proton wavefunction as a quantum object under discrete variable representation.

Contents
口試委員會審定書ii
Acknowledgements iii
摘要v
Abstract vii
I Introduction 1
1 Reactions with Post-Transition-State Bifurcation (PTSB) 3
II Literature Review 9
2 Multidimensional Potential Energy Hypersurface 11
3 Born–Oppenheimer Molecular Dynamics (BOMD) 15
III Methodology 21
4 Construction of 2D-PESs of PTSBs 23
4.1 2D-PES of Special Case: Symmetric PTSBs . . . . . . . . . . . . . . . . 25
4.2 2D-PES of General Cases: Asymmetric PTSBs . . . . . . . . . . . . . . 26
5 BOMD Trajectories and Their Mapping to 2D-PESs 29
IV Results and Discussion 31
6 Symmetric and Asymmetric PTSBs 33
6.1 Symmetric PTSB: H3CO Isomerization . . . . . . . . . . . . . . . . . . 33
6.2 Asymmetric PTSBs: Net C C Insertion for Nitrenes . . . . . . . . . . . 36
V Other Research Projects 41
7 Proton-Coupled Electron Transfer (PCET) 43
7.1 Classification of Concerted PCET: HAT and EPT . . . . . . . . . . . . . 44
7.2 Vibronic Coupling of a Model System . . . . . . . . . . . . . . . . . . . 46
8 Review of Anharmonicity Constants 51
8.1 Rayleigh-Schrodinger Perturbation Theory . . . . . . . . . . . . . . . . . 52
8.2 Choose the Zeroth-Order Hamiltonian as Harmonic Oscillator . . . . . . 55
VI Appendix 67
A Cartesian Coordinate for the Stationary Structures in Methoxyl Radical Isomerization
69
A.0.1 Transition-State Structures . . . . . . . . . . . . . . . . . . . . . 69
A.0.2 Local Minimum Structures . . . . . . . . . . . . . . . . . . . . . 70
B Time Evolution of Dynamic Trajectories 73
C Superimposition of Two Conformers: Rotation 77
D Electronic Structure Theory 83
D.1 Derivation of Hartree-Fock Approximation . . . . . . . . . . . . . . . . 84
D.2 Correlation Energy and Ground State Methods . . . . . . . . . . . . . . . 90
E Proton Wavefunction under Discrete Variable Representation 97
Bibliography 99
Copy of Permission 115

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