近年來大家關注在金奈米顆粒的研究由於金奈米顆粒有很高的吸收光效率，這個性質可以被應用在工程、能源科技以及生物科系上。舉例來說，金的奈米棒可以用來放在腫瘤細胞上面，接著用放射線去照射該腫瘤細胞，這個腫瘤細胞便可被摧毀。既然金的奈米結構的吸收效應是廣為人知且熱門的，於是我們便來探討這個性質以及如何使用結構來調變其吸收光譜。
我們使用時間相關的密度泛函理論來計算金奈米結構的吸收光譜。在本篇論文中，我們計算了三種金的奈米結構：第一種是金的線性原子鏈，第二種是金面心立方001 表面的平面結構，第三種是利用金面心立方結構堆疊出來的原子團。在本篇研究中我們討論了結構的形狀與結構的大小如何影響吸收光譜。此外，我們也討論了結構優化對於吸收光譜的影響。金的線性原子鏈是我們計算的結構中有最大吸收強度的結構，它的吸收峰僅有幾個較大的峰值，由以上2點可以判斷出金的線性原子鏈有集合電漿子的性質。

So many people paid close attention to Au nanoparticles in recent years because of their high optical absorption property. It can be applied in engineering, biotechnology, energy technology, etc. For example, gold nanorods can be put in breast tumor cells, then using irradiation to trigger it, the cells nally explode and be eliminated. Now that the optical absorption of Au nanostructures is so interesting, we would like to investigate properties of them.
The optical absorption spectra of Au nanostructures were calculated by using time evolution method which is based on time-dependent density functional theory. Here we present the spectra of three types of Au clusters; one is linear chain structures (Aun, n 9), another is planar structures from face-center cube (fcc) (001) plane, and the other is three-dimensional structures stacked by face-center crystal structure. In our study we discuss how cluster structure a ects absorption spectrum and the in uence of cluster size on absorption spectrum. Moreover, we also discuss the e ects of geometric optimization on absorption spectrum. The geometric shapes of these structures result in di erent absorption peaks and intensity. The linear chain clusters with three or more atoms can be regarded as collective plasmon because their absorption spectra have large absorption peaks and are dominated by those peaks.

1 Introduction 6
2 Theoretical Backgrounds and Computational Method 8
2.1 Hartree and Hartree-Folk Equations . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Hartree equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Deriving Hartree-Fock equations from Hartree equation . . . . . . 10
2.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 The Hohenberg-Kohn Theorem . . . . . . . . . . . . . . . . . . . 15
2.2.2 Extension to Ground State Energy Functional . . . . . . . . . . . 19
2.2.3 The Kohn-Sham Scheme and The Kohn-Sham equations . . . . . 22
2.3 Time-Dependent Density Functional Theory . . . . . . . . . . . . . . . . 26
2.3.1 The Runge-Gross Theorem . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Time-Dependent Kohn-Sham equations . . . . . . . . . . . . . . . 29
2.3.3 Time Evolution Method . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Computational Package - Octopus . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Calculation details . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.2 Propagator for the time dependent Kohn-Sham equations . . . . . 34
2.4.3 Delta kick - Calculating an Absorption Spectrum . . . . . . . . . 35
3 Calculations of Absorption Spectra of Au clusters 37
3.1 Bond length and vibration frequency of Au diatomic molecule . . . . . . 38
3.2 Atomic structures of Au Nanostructures . . . . . . . . . . . . . . . . . . 39
3.3 Absorption spectra of single Au atom and Au dimer . . . . . . . . . . . . 41
3.4 The effects of cluster size on absorption spectra . . . . . . . . . . . . . . 42
3.5 The effects of geometric shape on absorption spectra . . . . . . . . . . . 47
4 Influence of structure optimizations on absorption spectra 50
4.1 Stability of linear, planar, three-dimensional Au nanostructures . . . . . 51
4.2 Absorption spectra of the optimized Au nanostructures . . . . . . . . . . 51
5 Summary 55
Bibliography 57

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