臺灣博碩士論文加值系統

在這篇論文裡，我們利用半經驗Pariser-Parr-Pople(PPP)模型，研究了任意光學向量奈米碳管的基態與激態。我們發現，一般認為的奈米碳管能隙與光學向量的關係，只在緊束縛近似下有效。如果將電子作用力納入考量，即使只用平均場近似，在奈米碳管能帶中滿足mod(m-n,3)=0之偶然簡併將會消失，唯有扶手型奈米碳管得以豁免。

我們分析了奈米碳管簡並存在的對稱條件，也利用PPP模型做了Hartree-Fock的能帶計算。對於半導體奈米碳管，其結果與文獻上之密度泛涵理論(DFT)計算相符；對於非扶手型金屬奈米碳管之零能隙的破壞，也做了計算上的確認。

最後我們利用包含了長距作用力的PPP漢米吞算子，進行中介激子理論計算，以研究半導體奈米碳管的電子電子關聯效應。在利用複螺旋對稱及週期性條件後，我們把單位晶格裡的碳原子數降到二，大大地加速了單組態作用計算。也確認了奈米碳管的低激態態量與其半徑及螺性的關係。

這篇論文大致組織如下，第一章我們引入了適用於移動對稱系統之Bloch理論，然後再研究石墨與奈米碳管的關係後，將其推廣至複螺旋對稱系統。我們在第二章介紹HF的計算結果及奈米碳管能帶的一般性質與結構。第三章推導了一些中介激子理論的公式以及實際的計算結果。最後在附錄則有用MATLAB語言撰寫的量子化學工具箱的程式碼，並附上一點介紹。

In the present thesis, we performed a systematic study on the ground and excited states of carbon nanotubes with arbitrary chiral vectors based on the semi-empirical Pariser-Parr-Pople (PPP) model.
We found that the generally accepted relationship about the band gaps of single-walled carbon nanotube (CNT) and the chiral vectors, $(m,n)$, is only valid in the tight-binding approximation. When the electron interaction, as present in the PPP Hamiltonian, is included even at the level of mean field approximation, the accidental degeneracies in the energy bands of CNTs satisfying the condition, mod(m-n, 3)=0, are removed completely except armchair CNTs.
We have analyze the symmetry requirement of the existence of degeneracy in CNTs. We have also calculated the their Hartree-Fock band structures using the PPP Hamiltonian. The agreements with DFT calculations for semiconducting CNTs in the literature are satifying. For metallic CNTs, the absence of zero band-gap is also justified computationally for non-armchair systems.
Finally, we investigated the effect of electron-electron correlation on the excitonic electronic structures of intrinsically semiconducting carbon nanotubes (CNTs) by using the intermediate exciton theory with the long-ranged Pariser-Parr-Pople Hamiltonian. By taking full advantage of double helical symmetry and periodic condition, we are able to reduce the number of carbon atoms in a unit cell to two to facilitate the single configuration interaction calculation tremendously. The dependence of the excitation energies of low-lying excited states on the diameters and helicity of CNT is studied.
This thesis is organized as follows. In chapter 1, we first introduced Bloch''s theorem for ordinary system with translational symmetry and establish the double helical symmetry by making a correlation between graphene and CNT. The results of HF calculation and some general properties of band structure of CNT are presented in chapter 2. Finally some formula for performing intermediate exciton calculation are derived and the results are given in chapter 3. The quantum chemistry toolbox with matlab developed for this thesis are summarized in appendix with some comments.

1 Symmetry and Degeneracy in Quasi-One-Dimensional Systems
1 1.1 Introduction 2
1.2 Bloch''s Theorem 2
1.2.1 Two Simple Examples 3
1.3 The geometry of CNT 5
1.3.1 PA in CNT 6
1.3.2 Screw symmetry and generalize Bloch''s theorem 7
1.4 The Criteria of Degeneracy 11
1.4.1 PA and graphite 12
1.4.2 CNT 14
2 Hartree-Fock Crystal Orbitals 17
2.1 Hartree-Fock approximation 18
2.2 Pariser-Parr-Pople Model 20
2.2.1 A Brief Summary 20
2.2.2 Electron-Hole Symmetry 21
2.3 PPP for Polymer (PPPP) 23
2.3.1 A Brief Summary 23
2.3.2 PPPP for CNT 24
2.4 Results and Discussions 27
2.4.1 General property 27
2.4.2 Zigzag and armchair CNTs 30
3 Excitonic Structures of CNTs 37
3.1 Configuration interaction 38
3.2 CI in polymer 39
3.2.1 CI in PPPP 39
3.2.2 Selection Rule 41
3.2.3 SCI in CNT 43
3.3 Results and Discussion 44
A Quantum Chemistry Toolbox for Matlab 45
A.1 HF Calculations for trans-butadiene 47
A.1.1 Unrestricted-Hartree-Fock method for an open shell system 50
A.2 Single Configuration Interaction 52
A.3 Vectorized SCI Matrix 55
A.4 Direct SCI calculations 56
A.5 HF Crystal Orbital Calculations 57
A.5.1 Hückel calculation 59
A.5.2 HF calculation 60
A.6 Exciton Calculations 66
A.6.1 CI calculation 66
A.7 Distance Matrix 69
A.8 Geometry of Single-Walled Carbon Nanotubes 72
A.9 Random Phase Approximation and HF Stability Matrix 77

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