具備正確漸近性質的密度泛函評定，以及在TMA-H+-H2O例子上的正交模式分析改進

The Benchmark of Density Functionals with Correct Asymptotic Behavior, and the Improvements for the Normal Mode Analysis in the Case of TMA-H+-H2O

蔡宸瑋(Chen-Wei Tsai)

指導教授 ： 蔡政達

共同指導教授 ： 郭哲來(Jer-Lai Kuo)

國立臺灣大學/理學院/物理研究所/碩士(2012年)

https://doi.org/10.6342/NTU.2012.02772

摘要

本篇論文主要由兩個部分所組成。第一個部分是有關密度泛函理論的發展：我們做了密度泛函理論中具備正確漸進性質的密度泛函的表現評定。漸進性質已經被知道在雷德堡激發（Rydberg excitation）或是在分子前端軌域能量的計算時很重要，當然我們的目標是要發展一個全域範圍準確的密度泛函理論，但是當我們尚未達到這個目標之前，我們可以階段性地先要求一個密度泛函在漸進性質方面表現正確。我們對於當今的兩種主流漸進性質校正方法：模擬漸進位能校正模式（asymptotically corrected model potential scheme）以及遠距離混成校正模式（long-range corrected hybrid scheme），比較了它們在各種化學性質上的表現。我們發現，採用遠距離混成校正模式的密度泛函，幾乎在我們所比較的所有化學性質上都有不錯的表現；然而採用模擬漸進位能校正模式的密度泛函，只有在雷德堡激發以及前端軌域能量的計算上有比較好的表現，而其他化學性質的預測都非常地差。

另外在評定密度泛函表現的過程中，我建立了一個名為「IP131 Full Set」的分子資料庫，這個資料庫包含了131個原子或分子的實驗座標，以及這131個原子或分子的實驗垂直游離能（vertical ionization potential）、131個由CCSD(T)所計算出來的垂直電子親和力（vertical electron affinity）、131個由CCSD(T)所計算出來的分子基本能量間隙（fundamental gap）和113個由CCSD(T)所計算出來的分子拆散能（atomization energy）。IP131 Full Set將可用在其他第一原理計算方法的性質比較，或是未來其他密度泛函理論發展時的係數擬合（parameter fitting）。

在論文的第二個部分，是有關利用低維度薛丁格方程式（Schrodinger equation）的數值解，去校正與改善「正交模式分析（normal mode analysis）」在TMA-H+-H2O這個叢集分子上的分析結果。正交模式分析是一個我們常用來分析力學系統振動行為的方法，然而正交模式分析為了讓它的分析過程簡單並且系統化可解，設立了一些假設；這些假設固然方便，但也因為如此讓正交模式分析並非適用於所有的情況。例如在我們的例子TMA-H+-H2O中，正交模式分析的某些分析結果就顯得不太正確。我們主要的解決方法，是將一個或兩個正交模式分析的振動模式，視為一維或是二維的量子系統，並且採用薛丁格方程式在這些低維度量子系統上的數值解，去取代原本不適當的正交模式分析結果，最後經過一連串的修正，去盡可能地得到與實驗紅外線光譜一致的結果。藉此過程，我們也更加了解正交模式分析的適用情況，以及未來再遇到相同問題時，我們可以採用的修正方法。

關鍵字

密度泛函理論 ； 漸進性質 ； 正交模式分析 ； 薛丁格方程式 ； 紅外線光譜

並列摘要

This thesis primarily consists of two parts. In the first one, we benchmarked the performance of the functionals in density functional theory which have the asymptotically correct behavior. We compare the performance of the two different asymptotically corrected schemes, the long-range corrected (LC) hybrid scheme and the asymptotically corrected (AC) model potential scheme, in a wide range of applications. These schemes aim at producing the correct asymptotic behavior, which is known to be important to the Rydberg excitations and the frontier orbital energies of molecules. Based on our investigation, we found that the LC hybrid scheme outperforms the AC model potential scheme almost on all the properties studied, and we conclude that the LC hybrid scheme provides a promising direction for the future development of new exchange-correlation functionals.

Besides, in the process of the benchmark, I established a molecular database ``IP131 full set', which consists of 131 experimental geometries of atoms and molecules, and their reference values of 131 experimental vertical ionization energies, 131 CCSD(T) calculated vertical electron affinities, 131 CCSD(T) calculated fundamental gaps and 113 CCSD(T) calculated atomization energies. The IP131 full set can be used in the benchmarks for other Ab Initio calculation methods, or in the parameter fitting in the development of density functionals in the future.

In the second part, we adopted the Schrodinger equation numerical solutions to improve the applicability of the normal mode analysis in the case of TMA-H+-H2O, for the motivation that the theoretical simulated infrared spectrum of TMA-H+-H2O given by the normal mode analysis is far from the experiment results. Normal mode analysis is a convenient and widely adopted method that people usually use it to analyse the vibrational motions of multi-body mechanical systems, both classical and quantum. However, normal mode analysis is a approximate method and its assumptions, which make the analysis procedures simple and systematically solvable, also make the normal mode analysis not applicable to some vibrational motions in some mechanical systems. Our main modification method in this work is to regard the vibrational normal modes as some low dimensional quantum systems and adopt the Schrodinger equation numerical solutions in these quantum systems to replace the results from the normal mode analysis. By doing so, we can not only get a closer result to the experimental infrared spectrum, but also have a deeper understanding of the capability and applicability of normal mode analysis. Once in the future we encounter same problems when using the normal mode analysis in different systems, we may again adopt the procedures developed here to make it better.

並列關鍵字

density functional theory ； asymptotic behavior ； normal mode analysis ； Schrodinger equation ； infrared spectrum

參考文獻

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