指導教授：鄭原忠臺灣大學：化學研究所劉奕賢Liu, Yi-HsienYi-HsienLiu2014-11-252018-07-102014-11-252018-07-102014http://ntur.lib.ntu.edu.tw//handle/246246/261332由勢能面錐形交叉產生的幾何位向，可以理解為對絕熱矢量叢所造成的結構改變。但在本篇論文中，吾人證明了在實漢米頓算符的系統中，此絕熱矢量叢結構改變僅發生在實矢量叢中，其對應的複矢量叢結構仍是顯然的。由於原子核波函數是一個複函數，因此只有複矢量叢是重要的。故在實漢米頓算符的系統中，拓樸不變量無法做為判斷勢能面錐形交叉是否存在的工具。此外，由於錐形交叉的維度在複微擾下會下降，因此對大部份實漢米頓算符的錐形交叉點而言，錐形交叉會因複微擾而自鄰近的區域消失。 吾人亦考慮了A3分子做為例子，其中A為鹼金屬元素。從此例中，吾人證明除了一些特殊的原子核位置外，勢能面錐形交叉會因外加一個無窮小的磁場而消失。與此同時，吾人亦證明A3分子的半整數準旋轉量子數可以用一個沒有錐形交叉的複漢米頓算符解釋。因此，準旋轉自由度的半整數量子化現象並不能做為勢能面錐形交叉存在的證據。 在複漢米噸算符的系統中，吾人對所有原子核坐標流形中的二維曲面S定義了一個交叉量，記為CN(S)。這個量可以被用來偵測曲面上矢量叢的結構變化，因此給出了此曲面內錐形交叉個數的下限。另一方面，吾人亦證明了一個曲面的交叉量與錐形交叉的關係，可以類比為積分型高斯定律與帶電粒子的關係。此外，吾人亦提供了一個簡單的系統，以做為矢量叢結構改變造成準旋轉光譜改變的例證。 本文得到的另一個結果是，吾人一般化了由S. C. Althorpe 所提出來的計算方法。此方法提出之目的為將幾何位相的變化納入討論，但僅試用於有限的系統，而吾人的結果則將其推廣至任意帶有實漢米頓算符的系統。The geometric phase induced by conical intersection can be understood as structural change of adiabatic state line bundle. However, we proved that for real Hamiltonian, this structural change only occur for real line bundle and the corresponding complex line bundle is still trivial. Since nuclear wavefunction is a complex-valued function (or section), only complex line bundle is relevant. Hence we conclude that the existence of conical intersection cannot be determined by topological invariants for real Hamiltonian. We also showed that due to reduction of seam dimension, conical intersection disappear from the nearby region for most conical intersection points of original real Hamiltonian under complex perturbation. As an example, we consider the A3 molecule with A being an alkali atom. We showed that the conical intersection disappears under arbitrarily weak magnetic field except at some specific nuclear position. The half-integer pseudo-rotational quantum number, which is believed to be an evidence of existence of conical intersection, is also explained using a perturbed complex Hamiltonian without conical intersection. Hence half-integer quantization of pseudo-rotational degrees of freedom does not prove the existence of conical intersection. For systems with complex Hamiltonian, a quantity called conical number is defined for 2-surface in nuclear manifold Mnu, denoted by CN(S). This quantity can be used to determine whether structural change occurred around the given surface or not, thus CN(S) gives a lower bound of conical intersections inside S. We also showed that the relation between this number and conical intersection is an analog of integral Gauss’ law and charged particles, and an example of how this structural change affects the pseudorotational spectrum is given. Another result is that the calculation method proposed by S. C. Althorpe to include geometric phase[38][39] is generalized to arbitrary system with real Hamiltonian.口試委員會審定書i 中文摘要ii 英文摘要iii 1 Introduction 1 1.1 Break Down of Born-Oppenheimer Approximation . . . . . . . . . . . . 1 1.2 Simulation Methods for Non-adiabatic Dynamics . . . . . . . . . . . . . 3 1.2.1 Fewest Switch Surface Hopping . . . . . . . . . . . . . . . . . . 3 1.2.2 Ehrenfest Mean Field . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Path Integral for Non-Adiabatic Dynamics . . . . . . . . . . . . 6 1.2.4 Initial Value Representation of Semiclassical Theory . . . . . . . 8 1.2.5 Mixed Quantum-Classical Liouville Equation . . . . . . . . . . . 10 1.3 Conical Intersection and Geometric Phase . . . . . . . . . . . . . . . . . 11 1.4 Geometric Phase and Topology . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.2 Vector Bundles and Sections . . . . . . . . . . . . . . . . . . . . 18 1.5.3 Covering Spaces and First Fundamental Group . . . . . . . . . . 26 1.6 Topology and Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . 33 1.7 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 Path Integral Formula for Non-Adiabatic Dynamics 37 2.1 Derivation of Path Integral Formula . . . . . . . . . . . . . . . . . . . . 39 2.2 Path Integral Formula with Conical Intersection . . . . . . . . . . . . . . 42 3 Non-Existence of Topological Invariants for Real Hamiltonian 47 3.1 Triviality of Complex Vector Bundles for Real Hamiltonian . . . . . . . . 48 3.2 Disappearance of Conical Intersection – A Model Study . . . . . . . . . 51 3.3 Why Conical Intersection Disappear? – A Brief Discussion . . . . . . . . 59 4 Topological Invariants for Complex Hamiltonian – a Simple Application 63 4.1 Detecting Conical Intersection . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Model Hamiltonian Near Conical Intersection . . . . . . . . . . . . . . . 66 4.3 Conical Number for General Hamiltonian . . . . . . . . . . . . . . . . . 68 4.4 Physical Analog of Conical Number . . . . . . . . . . . . . . . . . . . . 75 4.5 Conical Number in Higher Dimension . . . . . . . . . . . . . . . . . . . 78 4.6 Half-Integer Pseudo-Rotational Quantum Number Induced by Conical Intersection and its Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Conclusion 87 A Proof of Theorem 3.1.1 99 B Proof of Theorem 3.1.2 105 C Proof of Theorem 4.5.1 115 D Solving Y ( ; ) 1171255960 bytesapplication/pdf論文公開時間：2014/03/09論文使用權限：同意有償授權(權利金給回饋學校)錐形交叉幾何位相拓樸不變量thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/261332/1/ntu-103-R99223129-1.pdf