本研究使用機器學習方法，以小系統的密度泛函理論計算數據訓練機器學習模型，再以得到的模型預測大系統的能量與結構，最終由反應物與生成物的預測能量計算反應能。我們以此方法研究共價有機骨架(covalent organic frameworks,COF) 合成反應與產氫反應。對於COF 結構單元的合成反應，我們發展出以隨機微擾結構作為訓練數據的方法，隨機小幅度擾動平衡結構內的原子，再以密度泛函理論計算得到此微擾結構的能量與各原子受力。我們發展的方法可以控制數據量，並較好的對位能表面取樣，使得以此為訓練資料的機器學習模型預測效果更好，其中最精準的模型預測能量與密度泛函理論計算的平均誤差約小於15 meV/atom。對於純金屬表面的產氫催化反應，我們以1×1 與2×2 的小尺寸表面晶胞(含純表面、0.25ML 與1ML 吸附氫原子表面) 的隨機微擾結構數據為訓練資料，訓練得到的機器學習模型能準確預測3×3 與4×4 大尺寸表面晶胞的能量與結構。由大尺寸表面晶胞的預測能量計算得出的氫原子吸附能與密度泛函理論計算的結果相當，其誤差幾乎都小於100 meV。本研究實現以微小計算量的小系統結構數據，準確預測較大系統的能量與結構，且能應用預測結果於計算反應能。

This study employed machine learning methods. We utilized density functional theory calculation to generate database for small systems. The databases was used to train machine learning models, which predicted the energy and structure of larger systems. Ultimately, the reaction energies could be calculated, based on the predicted energies of reactants and products. We used this approach to investigate the synthesis reactions of covalent organic frameworks (COF) and hydrogen production reactions. For the synthesis reactions of COF structural units, We developed a method adopting randomly perturbed structures as training data, wherein small random perturbations were introduced to the equilibrium atomic positions, and the energy and forces of these perturbed structures were calculated using density functional theory. This method allows us to controll the amount of data and improve sampling of potential energy surfaces, resulting in better predictive performance of machine learning models trained on this data. The most accurate model achieves an average error of less than 15 meV/atom when predicting the energy of density functional theory calculations. We exploited randomly perturbed structures of 1×1 and 2×2 small-sized surface cells (including pure surface, 0.25 ML and 1 ML hydrogen-absorbed surfaces) as training data to predict hydrogen catalysis reaction on the pure metal surface, and the obtained machine learning model can accurately predict the energy and structure of larger 3×3 and 4×4 surface cells by the trained models are accurately matched the results from DFT calculation. The predicted hydrogen adsorption energies for the large-sized surface cells closely align with the result of density functional theory calculations, with errors consistently below 100 meV. This research demonstrates the ability to accurately predict the energy and structure of larger systems using computational data from small systems and apply these predictions to calculate reaction energies.

論文審定書. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
英文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
第一章緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 有機反應與催化反應. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 有機反應. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 催化反應. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 密度泛函理論與應用. . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 密度泛函理論. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 密度泛函理論的應用. . . . . . . . . . . . . . . . . . . . . . . 3
1.3 機器學習方法與應用. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 機器學習方法. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 機器學習方法於材料科學的應用. . . . . . . . . . . . . . . . 5
1.3.3 機器學習方法於化學反應的應用. . . . . . . . . . . . . . . . 7
第二章計算理論分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Born-Oppenheimer 近似. . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 單電子與多電子系統. . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Born-Oppenheimer 近似. . . . . . . . . . . . . . . . . . . . . . 11
2.2 Hartree-Fock 近似. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Hartree 近似. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Hartree-Fock 近似. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 密度泛函理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Hohenberg-Kohn 理論. . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Kohn–Sham 方程式. . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 機器學習. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 人工神經網路與深度神經網路. . . . . . . . . . . . . . . . . . 18
2.4.2 卷積神經網路. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
第三章計算方法與結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1 計算套件與設定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 VASP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 SchNetPack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 COF 結構單元的合成反應. . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 以結構優化過程數據為訓練資料的模型. . . . . . . . . . . . 28
3.2.2 以隨機微擾結構數據為訓練資料的模型. . . . . . . . . . . . 30
3.2.3 以分子片段結構數據為訓練資料的模型. . . . . . . . . . . . 33
3.2.4 預測反應的結果. . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 純金屬表面的產氫催化反應. . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 以最小表面晶胞數據為訓練資料的模型. . . . . . . . . . . . 43
3.3.2 以不同氫吸附率表面晶胞數據為訓練資料的模型. . . . . . . 47
3.3.3 以表面晶胞兩層原子隨機微擾的數據為訓練資料的模型. . . 48
3.3.4 預測反應的結果. . . . . . . . . . . . . . . . . . . . . . . . . . 50
第四章結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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