一個在可變調格點下使用內插尺度函數的多重解析度新方法，可用於第一原理計算。我們也以二次微分的內插尺度函數解帕松微分方程，也得到了很好的計算結果。此法比碎波簡單，而且完全實現快速碎波轉換，所以也適合頻繁更新之電荷系統，像是原子、分子及固態電子結構的第一原理計算。雖然這方法可簡單的實現在實空間的可變調變格點法，格點數卻比起直接使用實空間少很多。這個簡單而有效的方法，提供了在實空間與碎波之外的選擇。另外，此法將不同解析度的空間分開處理，所以容易被平行化。

A new multiresolution scheme based on interpolating scaling function(ISF) on adaptive gridding(AG) shows promising in the first principle calculation. We also use ISFs on solving Poisson equation(PE), and find good approximations on the expansions of the second derivatives of ISFs. It is simpler than the wavelet scheme and fully implements the fast wavelet transformation so that the method is very suitable to problems with frequently updating charge density such as the first-principle calculation in electronic structures in atoms, molecules, and solids.
Although the scheme is similar to the AG scheme on real space, the ISFs can represent fields more effectively and it needs less grids than the scheme of real space does. This simple and effective method provides an alternative to both the real space and the wavelet methods in the first principle calculation. Also, The method can be easily parallelized due to the block structure of the grid layout.

第一章 前言..............................................................1

第二章 柯恩－沈方程..............................................6
2-1：柯恩-沈方程....................................................6
2-2：平面波...........................................................12
2-2-1：糕模勢.......................................................14
2-2-2：線性綴加平面波法...................................15
2-2-3：線性原子球軌道法...................................17
2-3：碎波......................................................... .....18
2-3-1：多重解析度...............................................28
2-3-2：基底函數選擇...........................................21
2-3-3：場值轉換...................................................22

第三章 內插尺度函數與第一原理計算................26
3-1： 特色..............................................................26
3-2： 製造內插尺度函數......................................31
3-2-1： 原始函數..................................................31
3-2-2： 一次微分函數.................. ........................41
3-2-3： 二次微分函數...........................................41
3-3： 多重解析度...................................................43
3-4： 第一原理計算...............................................47
3-4-1： 基底函數安排..................... ......................51
3-4-2：計算能量.....................................................54
3-4-3：求波函數解.................................................55
第四章 帕松微分方程.............................................63
4-1： 理論...............................................................63
4-1-1： 解庫倫位勢................................................63
4-1-2： 以內插尺度函數求位勢............................68
4-2： 二次微分圖形................................................69
4-2-1： 平滑化再微分............................................69
4-2-2：解方程式取二次微分展開值.....................73
4-2-3： 多重解析度解位勢....................................75
4-3： 實例................................................................78

第五章 成果與討論..................................................85
5-1： 系統安排及初始條件....................................85
5-2： 結果................................................................87
5-3： 結論................................................................90

參考文獻...................................................................93

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