應用於數學化學及量子化學之演算法及程式開發

Abstract

開發新的計算科學方法時，通常需要大量及繁瑣的人工處理，包含演算法推導、程式實作以及測試。現在這些人工處理通常可以被高階的演算法及程式取代。一個成功及有名的例子是由 Hirata 教授所發展的耦合簇及高階微擾理論方法之自動推導及程式實作。經由此程式的幫助，能夠減少人工程式實作上的錯誤。自動化的測試在開發計算科學方法上也有其重要性，能夠提昇生產力同時也能降低發生錯誤的機會。此論文探討在我博士班學習過程中所開發設計之應用於數學化學及量子化學上的兩個自動化系統。此兩系統包含：密度泛函緊束方法之參數調校自動化系統，以及苯系統之 Clar 覆蓋多項式的自動計算及推導系統。

第一個自動化系統，密度泛函緊束方法之參數調校自動化工具箱---其縮寫為 ADPT---提供了一系列可自動化產生密度泛函緊束方法之參數的工具。這系統的主要目標是自動產生可重現使用者選擇的特定從第一原理計或是實驗取得之物理與化學性質的密度泛函緊束方法之參數。除此之外，此工具箱也提供了數個可以減少人工處理的工具，包含自動產生參數檔，以及自動參數檔測試之工具。在此論文中，我測試了數個使用此自動化系統所最佳化之參數，這些參數都可精確地重現由泛函理論所計算之物理化學性質。在此自動化系統中的主要演算法是基於一個名為群粒子最佳化的演算法，此演算法已經被應用於 很多其他領域。此自動化系統也提供出應用程式開發界面讓未來的開發者能夠容易地整合及應用此系統。

第二個自動化系統，是用於計算以Clar 覆蓋多項式及推導其閉型解。Clar 結構可以視為是一般化的凱庫勒結構，提供有機化學家一個分析及預測有機分子穩定度及反應性的方法。計算 Clar 覆蓋多項式是一個非常複雜的工作，需要將所有其芳香系統中所有的 Clar 結構。手動方式計算中型大型分子的 Clar 覆蓋多項式實際上是不可能的。本論文中，我開發一個平行化的電腦程式，用於有效率地自動計算 Clar 覆蓋多項式。此程式為第一個發表之計算 Clar 覆蓋多項式的程式。在此論文中，我也開發了令一個用於自動推導 Clar 覆蓋多項式的圖形化程式。此程式可以讓推導 Clar 覆蓋多項式之閉型解變得相當容易及快速。本論文中，此兩個程式被用於非常多類型的芳香系統，其大多數之閉型解也在此論文中提出及討論。

Developing new computational methods often requires large amount of manual work, including derivation of theoretical formulas, implementation of computer code, and performing extensive benchmarking and testing. Nowadays, these manual tasks can often be replaced or assisted by advanced computer programs. A well-known and successful example is the automatic derivation and implementation of perturbation theories and coupled-cluster theories developed by Hirata. The assistance from computer algorithms and programs can minimize human mistakes in the manual coding process. Automation of testing processes can also increase the productivity and reduce human errors. In my thesis I discuss automatic computer environments designed during my Ph.D. program to assist in time-consuming and error-prone quantum chemical tasks: automatic parameterization of the density-functional tighting-binding (DFTB) method and automatic determination and derivation of Clar covering polynomials of benzenoids.

The first presented here automatic computer environment---further referred to as the automatic DFTB paramtrization toolkit or shorty ADPT---is a set of tools for automatic parameterization of the DFTB method. The ADPT toolkit was designed to generate DFTB parameter files able to reproduce selected chemical and physical properties obtained either from ab initio calculations or from experiments. The parameter optimization process is fully automatized. In addition, the ADPT provides also tools for optimizing and testing the parameter files in an automatic fashion. Application of the designed environment to several challenging parameterization tasks show that the ADPT toolkit can reproduce selected chemical and physical properties of molecular and crystal systems with reasonable accuracy in comparison with the density-functional theory (DFT) data. The ADPT optimization tool, named PSOSKOptimizer, uses a population-based global optimization algorithm, called particle swarm optimization or shortly PSO, which has been previously applied in various global optimization problems. The computer programs developed in this thesis also provide the application programming interfaces (APIs) that can be further integrated or extended by future developers.

The second presented here automatic environment has been designed to computing Clar covering polynomials of benzenoid systems. Clar covers can be thought as a generalization of the usual Kekule structures used by organic chemists to analyze and predict stability and reactivity of various polycyclic compounds. Computation of the Clar covering polynomials of benzenoids involves huge amount of manual tasks related to enumeration of all possible Kekule structures with a given number of Clar aromatic sextets. For medium or large systems, finding the Clar covering polynomials in a pencil-and-paper fashion is practically infeasible. In this thesis, an efficient, parallelized computer program, called ZZCalculator, has been developed for automatic determination of the Clar covering polynomials for a large class of benzenoids. ZZCalculator is the first available computer code for computing the Clar covering polynomials. With the help of this automatic tool, closed-form formulas of the Clar covering polynomials of different classes of benzenoids have been derived. A Web interface, accessible via http://qcl.ac.nctu.edu.tw/zzpolynomial, designed for instant computation of the Clar covering polynomials of benzenoids has also been developed. Identification of the recurrence relations for the Clar covering polynomials of benzenoids is crucial for formal derivation of the closed-form formulas of the Clar covering polynomials. To improve this cumbersome process, in this thesis a proof-oriented graphical program, called ZZDecomposer has been presented. It provides an easy, elegant, and automatic way of tracking all benzenoid substructures appearing in the process of decomposition needed to discover the underlying recurrence relations. With the assistance of ZZDecomposer, finding and deriving the closed-form formulas of the Clar covering polynomials of benzenoids are made possible and easy. As an application of the ZZDecomposer, the closed-form formulas for the Clar covering polynomials of many different classes of benzenoids have been formally derived in this study.