普适性能量泛函的严格形式至今还不清楚，特别是动量空间的泛函形式

论文标题:密度泛函理论中分子体系的界限研究
Studies on the Bounds to Density Functionals for Molecules in Density Functional Theory
论文作者 田国才
论文导师 李国宝；陶建民,论文学位 硕士,论文专业 物理化学
论文单位 云南师范大学,点击次数 100,论文页数 87页File Size2939k
2002-05-01论文网 http://www.lw23.com/lunwen_200777212/ 界限研究;变分原理;信息论方法;信息熵;坐标矩;坐标和动量期望值;密度泛函理论
Study on the bounds,Variational Principle,Information Theory Approaches,Information Entropy,Coordinate Moments,Radial Expectation Values and Momentum Expectation Values,Density Functional theory
多粒子体系传统的处理方法是通过求解Schrdinger方程，用求解得到的波函数描述原子、分子的性质，在第一章中我们简述了多粒子体系的一些常用方法如：Hartree方法Hartree-Fock方法等，并比较了各种方法的优缺点，这些方法通常称为波函数理论。我们知道波函数的物理意义至今仍然不明确，而且实验不可观测，再者因多粒子体系间的相互作用比较复杂，求解Schrdinger方程就显得比较困难了。尽管随着现代计算技术的发展，多体问题Schrdinger方程的精确解已经得到并成功地应用于预测原子、分子、固体和表面的性质，但是所有这些基于波函数理论方法在计算上都是非常昂贵和费时的，特别对于大体系更是如此。 对于体系的基态，Hohenberg，Kohn和Sham他们证明了多电子体系基态的所有性质都是电子密度的唯一泛函，即体系基态的电子密度分布可以给出体系的所有信息。他们提出将电子密度作为基本变量来研究多电子体系的结构和性质，这样使复杂的3N维多电子波函数及其对应的薛定谔方程转化为简单的3维可通过衍射实验测定的单粒子电子密度及相关理论。从此用电子密度来处理多电子体系的密度泛函理论(DFT)已经广泛地应用于凝聚态物理，并且更多地应用与预测化学物质的性质、化学反应的趋势和速率以及生物大分子的性质，这些性质的预测将对新材料、新化学物质和新药物的设计提供帮助。基于在密度泛函发展方面作出的杰出贡献，1998年Nobel化学奖授予了美国学者Walter Kohn。在第二章中我们对密度泛函理论基础及其推广做一简短叙述，将总能量的一些组成项定义为电子密度的泛函，并初步讨论了这些泛函的性质。 然而在密度泛函理论(DFT)的研究过程中，存在着巨大挑战，普适性能量泛函的严格形式至今还不清楚，特别是动量空间的泛函形式。因此人们需要作的主要工作是建立和改进以电子密度表示的动能和交换相关能泛函。建立和改进动能和交换相关能泛函已经成为当今密度泛函理论研究的焦点。在这一研究中有两个主要的研究方向：其一是采用各种近似方法来建立和改进动能和交换相关能的近似泛函。如，局域密度近似(LDA)、梯度展开近似(GEA)、广义梯度展开近似(GGA)等。这些近似方法我们在第三章中进行了简单阐述。其二是对动能和交换相关能泛函的界限进行研究。界限研究是一种非常有趣和重要的方法，通过界限研究可以为寻找近似泛函和普适性泛函指明方向，为各种近似方法的精度提供判断依据，为数值拟合近似泛函以改进或简化计算方法提供信息，为建立理论和实验的联系而进行理论解释和预测。后者是本文研究的重点，主要内容放在论文的第四 云南师范大学硕士学位论文 摘 要章。在第四章中，我们在综述前人工作的基础上，利用信息论方法，以变分法为主线贯穿全文，系统地研究了密度泛函理论中的界限问题，并将所得的一些有趣的结果首次推广到了分子体系，对分子体系进行了研究，得到的主要结果如下： 1．原子体系的信息嫡的界限 2．原于体系的信息嫡与总电于数、动能、总能量之间的关系 3．经典库仑能、交换能的上下界，并根据所得结果得出了经典库仑能、交换能 的近似泛函 4．坐标矩之间的关系，并首次将一些结果推广到了分子体系 5．坐标期望值和动量期望值之间的关系，根据所得结果得出了穴－’＼＜p”’＞和 印＞的近似泛函，并首次将所得结果推广到了分子体系 在这个研究中，我们发现信息论方法在密度泛函理论的界限研究中非常有用。应用信息论方法来研究原子、分子体系，得出的结果比现有的其他方法都要漂亮得多，所得的结果大部分都可以推广到分子体系。对分于体系的研究发现对只含有O－轨道的分子，因分子的对称性高，由信息论方法得出的结果与原子体系一样好，对于含多重键的分子因对称性差，得出的结果不如原于体系好．我们期待着一些更新，更严格的各物理量之间关系的导出。我们的工作进一步说明在密度泛函理论的研究中，通过一些容易获得的物理量的知识去估计一些重要的物理量的过程中，信息论是一种尤为有用的方法，由此得出的结果更加具有普适性。文中普遍使用了原子单位。
The commonly-used approaches applied to many-body system are solving the Schrodinger equation and describe the electronic structures and properties of atoms, molecules and solids with the wavefunction that obtained by solving the Schrodinger equation of the system. In chapter one, we give a brief review of various approaches applied to the many-body system, such as Hartree Method、Hartree-Fock Method etc. The advantages and the disadvantages of these methods are reviewed. All of these methods are based on wavefunction, so it always called wavefunction theory. Although accurate solutions of the Schrodinger equation for the system have been found and successfully used to predict the structures and properties of atoms, molecules and solids with the development of modern computer technology. However, the physical significance of the wavefunction is still not clearly and definitely up to now as we know and it cannot be measured directly by experiments. Moreover, it is difficult to solve the Shrodinger equation of many-body system and the wavefunction theories are computationally expensive, especially for large system, since the interaction of many-body system is complicated.For ground states, Hohenberg, Kohn and Sham proved that all ground state properties of the system is a unique functional with electron density. In other words, Electron density determines all ground state properties of the system. They made a suggestion that ascribed the electron density as a basic variable to study the electronic structure and properties of many-body system. This allows one to replace the complicated 3N-dimensional wavefunction and the associated Schrddinger equation by the 3-dimensional single particle density which can be measured from diffraction experiments and readily visualized and its associated calculational scheme. Since then, The density functional theory (DFT) has been widely-used in condensed-matter physics and increasingly applied to chemistry for the prediction of these properties, chemical reaction trend and rate, and the properties of biologically interesting molecules. The prediction of these properties will contribute to design of new materials, chemicals, and pharmaceuticals. With the contributions of development for the density functional theory, the 1998 Nobel Prize was awarded to the American scholar Walter Kohn. In chapter Two, we give a concise description of the density functional theory, various ingredientsof the total energy are defined as functional of the electron density and some formal properties of these functionals are discussed.However, In the studies of the density functional theory, big challenges remain, the exact and rigorous form of the universal energy functional has still not been found, especially, in the momentum space so far. Largely having to do with the need to construct or improve functional of the electron density. Thus Construction or improvement of exchange-correlation functionals has been the focus of the density functional theory. There are two major directions in this domain: one is construction or improvement of approximations the kinetic and exchange-correlation energy functionals by various approximate methods. Such as local density approximation (LDA), local spin density approximation (LDA), general expansion approximation (GEA) and general gradient approximation(GGA) etc. Various approximate methods and their application in atom and molecules are reviewed briefly and descripted concisely in the chapter 3. The other is study on the bounds to the kinetic and exchange-correlation energy functionals. Studies on the bounds to density functional are very interesting and important approaches in density functional theory (DFT), and they could indicate the directions and provide much reliable information for finding or constructing the approximate and rigorous forms of the energy functional which could simplify and improve the computational methods. These are the major project we have done in present thesis. Based on the brief review of the works have been done by previ