非線性光學性質的應用很廣，而有機分子晶體基於其分子合成的多樣性及可調控性，成為材料設計有利的種類。我們針對有機分子晶體的零頻χ(2)　做一系列的探討,而希望建立系統化研究的經驗及試圖理解該材料產生SHG的機制。
    首先針對廣為研究的尿素我們分別進行單分子β與晶體χ(2)　的預測。針對各種計算參數做了詳細的收斂性測試，以建立利用平面波贗方法進行此類材料研究的經驗。除了一般性之空軌域數K採樣密度截止動能值外，我們發現計算分子所取的超晶胞大小是值得注意的。在用了能隙修正來彌補Kohn-Sham方法在能隙上的差距之後，可以得到與實驗吻合度不錯的χ(2)　値。
    基於計算尿素的經驗，我們展開一系列有機分子晶體其分子β與晶體χ(2)　的比較分析，為的是檢驗分子晶體之χ(2)　只要是由分子β張量加成而得這樣的說法是否普遍適用.。在十個進行計算的分子晶體(COANP、mNA、NPP、5NU、POM、MHBA、MDNB、PNP、L-PCA、FMA)NLO材料，我們發現大部分經體內整體分子的有效β値都有比個別分子之β加成大,即分子堆積成晶體含有β增強的效應。我們用能帶解析方法分析這十種晶體的χ(2)，發現對χ(2)　有重要貢獻的軌域除了是繼承自分子對β貢獻者外，晶體在未佔據態高能區，有其特有的新貢獻。這至少提供了增強效應可能來自何處的一種機制的解釋。
    有鑑於能隙修正在我們所採用的SUM-OVER-STATES方法要得到盡可能正確的χ(2)　値是頗重要的這個事實，我們也探討比較了幾種不同能隙修正方法下的結果。

The application of Nonlinear Optical Properties of Organic Molecular Crystals is very broad. Due to the multiplicity and adjustability of their molecules, organic molecular crystals become advantageous to materials design. In this thesis we focus on the zero-frequencyχ(2)of organic molecular crystals and do a series of research in order to build up systematic experience and to understand the mechanism for second harmonic generation(SHG) in these kind of materials.
  We first focus on widely studied urea and perform the predictions of β andχ(2) in single molecule and crystal respectively. The convergency of parameters in calculation has been carefully tested in order to build up the research experience in using plane wave pseudopotentials method to study these kinds of materials. We find the size of the selected super cell in molecular calculation is crucial besides the general parameters in calculation, such as numbers of unoccupied orbitals, k-point sampling and cut-off energy. After applying band gap correction to fix the band gap discrepancy between Kohn-Sham method and experiments, one can obtainχ(2) which shows good agreement with experiments.
    According to calculating urea , we launch a serial compared analysis between Organic Molecular Crystalsβand χ(2)  to verify the search of  the molecular crystalsχ(2) equals to βtotal value in general. In ten process calculating morlecular crystals: COANP、mNA、NPP、5NU、POM、MHBA、MDNB、PNP、L-PCA、FMA)NLO material, we realized most of molecular valuableβin crystals will be larger than the sum of each molecular, that is enhancing effects of calculating crystals. We could apply band resolve method to analyze the ten molecular χ(2), and we get one result contributing orbitals not only inherit from molecular but also contribute unique and new in unoccupied states area of high energy。 The above-mentioned proof enhancing effects derive from others systems. 
In view of band gap correction, we adapted the SUM-OVER-STATES  method to obtain a closely correctχ(2)　; It is a very important point of the research. We probe into various kinds of different band gap correction method results.

第一章 背景介紹 ....1
一．有機物在非線性光電材料的重要性……..1
二．第一原理計算與分析方法……..2
（一）以電腦模擬計算非線性光學性質的原因……..2
（二）方法理論……..2
　　　1．密度泛函理論……2
　　　2．平面波方法……3
　　　3．光學性質計算……5
　　　4．非線性光學計算…6
（三）分析工具(能帶解析方法)……7
三．研究內容……..8
參考文獻………10

第二章 尿素分子與晶體的非線性光學性質計算…..12
一．介紹……12
（一）動機……12
（二）計算策略……12
二．計算結果……12
（一）模型與設定參數介紹………12
（二）尿素晶體的氫鍵與能帶結構圖……13
（三）尿素晶體χ(2)計算結果與手冊χ(2)實驗值的比較…..16
（四）以分子軸為參考座標之分子β値與其在晶體內排列之效果…17
（五）尿素分子的β能帶解析圖與尿素晶體的χ(2)能帶解析圖…….21
（六）尿素分子與尿素晶體的SHG重要軌域…….23
三．收斂性測試的詳細探討……25
（一）為什麼相信CSD 模型?.....25
（二）計算尿素晶體與分子時採用LDA或是GGA?....26
（三）計算尿素晶體與分子多少空BAND數是足夠的?........26
（四）分子的BOX應該取多大?....29
四．結論...34
參考文獻…36

第三章 有機分子與晶體的非線性光學計算…….37
一．背景介紹……37
（一）動機……37
（二）計算策略……38
二．計算結果……43
三．結論………43
參考文獻……52

第四章 有機分子及晶體之密度泛函計算的能隙修正……53
一．介紹...53
（一）動機與介紹..53
（二）計算策略……55
二 修正結果的分析比較……55
（一）尿素晶體能隙修正結果比較………55
（二）第三章有機晶體GDFT能隙修正結果比較……58
（三）第三章有機晶體折射率擬合能隙修正結果比較…60
三 結論…61
參考文獻...61

附錄...64
附錄一 5-Nitrouracil(5NU)……64
附錄二 2-Cyclooctylamino-5nitropyridine(COANP)…79
附錄三 2-furyl-methacrylic anhydride(FMA)…97
附錄四 L-pyrrolidone-2-carboxylic Acid(L-PCA)…116
附錄五 C6H4(NO2)2,m-Dinitrobenzene(MDNB)…137
附錄六 3-methoxy-4-hydroxy-benzaldehyde(MHBA)…162
附錄七 m-Nitroaniline(mNA)…182
附錄八 N-(4-Nitrophenyl)-L-prolinol(NPP)…202
附錄九 2-(N-Prolinol)-5-n itropyridine(PNP)…221
附錄十 3-methy-4nitropyridine-1-oxide(POM)…239

第一章
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第二章
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2．K. Wu.,J. Li,Applied Physics A,76,427-431(2003)

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第三章
3.1Handbook of nonlinear optical crystals

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3.230 Appl.Phys. Lett.59,19-21(1991)

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3.624 Appl.Phys. Lett.50,486-488(1987)

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第四章
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[2]  John P. Perdew and Mel Levy, Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities, Phys. Rev. Lett. 51, 1884–1887 (1983)

[3]  R. W. Godby, M. Schlu"ter and L. J. Sham, Trends in self-energy operators and their corresponding exchange-correlation potentials, Phys. Rev. B 36, 6497–6500 (1987)

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[7]  L. Fritsche and Y. M. Gu, Band gaps in a generalized density-functional theory, Phys. Rev. B , Phys. Rev. B 48, 4250–4258 (1993)

[8]  L. Fritsche，Excited States and Electron-atom Scattering in Density Functional Theory，edited by E.K.U. Gross and R.M. Dreizler (Plenum Press, New York, 1995)

[9]  L. Fritsche，Reply to the comment by M Biagini on generalized density functional theory，Journal of Physics: Condensed Matter , Vol 8 , p.2237 (1996) 

[10]  I. N. Remediakis and Efthimios Kaxiras, Band-structure calculations for semiconductors within generalized-density-functional theory, Phys. Rev. B 59, 5536–5543 (1999)

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[13]  Mark S. Hybertsen and Steven G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Phys. Rev. B 34, 5390–5413 (1986)

[14]  A. Seidl, A. Gorling, P. Vogl, and J. A. Majewski，Generalized Kohn-Sham schemes and the band-gap problem，Phys. Rev. B , V53 , p.3764 (1995)

[15]  D. M. Bylander and Leonard Kleinman，Good semiconductor band gaps with a modified local-density approximation，Phys. Rev. B , V 41 , p.7868

[16]  L. Fritsche， Physica B , V173 , p.7 (1991)

[17] 陳冠雄,核修正下之廣義密度泛函理論的能隙修正及稀土鈣硼酸鹽晶體GdCa4O(BO3)3之光學性質計算,淡江大學物理學系,碩士論文

[18]  V.G.Dmitriev,Handbook of Nonlinear Optics Crystals