量子化学全册配套最完整精品课件1.ppt
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1、Syllabus Download the lectures from ftp:/ (user: wjliu; passwd: ccmebdf) References: 1. Chapters 1-4 of Modern Quantum Chemistry (A. Szabo and N. S. Ostlund) 2. Chapters 3-5 of Introduction to Computation Chemistry (F. Jensen) (preview Chapter 1 of Ref. 1 for linear algebra) 量子化学课考试方案量子化学课考试方案 考试原则:
2、主要考察对考试原则:主要考察对概念概念、原理原理、物理图像物理图像、计算方法计算方法的理解和运用的理解和运用 一、书面考试:一、书面考试:6月月10(星期二(星期二3-4节)(节)(60分分);); 二、上机实习:书面报告(二、上机实习:书面报告(15分分);); 三、文献综述:选择一个与量子化学有关的课题撰写小型评述(三、文献综述:选择一个与量子化学有关的课题撰写小型评述(4000字,字, 可用英文),要求主要文献中量子化学的内容占可用英文),要求主要文献中量子化学的内容占50以上,书面报告以上,书面报告 (10分,分,6月月24日前上交日前上交)应符合写作规范(引言、计算细节、正文、)应符
3、合写作规范(引言、计算细节、正文、 问题讨论、结论),并正确引用文献;问题讨论、结论),并正确引用文献; 四、口头报告:四、口头报告:5月月30、6月月4、6月月6进行口头(进行口头(15分分)。)。 鼓励课堂上提问题,如果提出好的问题,适当加分。鼓励课堂上提问题,如果提出好的问题,适当加分。 注:如条件允许,拟参加考试的旁听生可以随正式注册生上机实习。注:如条件允许,拟参加考试的旁听生可以随正式注册生上机实习。 Central contents Quantum vs. classical pictures HE What H ? What method? Basis Correlation
4、Relativity (in ) (in H) Mathematics: 2-by-2 matrix Models: H2 decoherence) There is so far no successful theory for decoherence. “do not wiseacre, just compute!” Uncertainty principle (Heisenberg, 1927) 1 | 2 ABABBA 1 |, | 22 xx xppx It is nonsense to speak of the momentum of a quantum particle at a
5、 given point. So is the orbit ! 2 | () |AAA (fluctuation) 21 EEE t The energy spacing E can only be resolved over time scales long compared with In other words, a long time MD simulation (observation) is to resolve energetically spacing states. Conversely, if a simulation (observation) lasts for a t
6、ypical time t, the corresponding energy resolution is of order . For a macroscopic system, the energy spacing E is usually much smaller than , such that only the level density, averaged over an energy interval , can be detected, reflected by a broad peak. /t /t /t/t 11 | |,| 22 d H AiA dt EA |) 2 |(
7、/ d AAE dt E Time is just a parameter! (mathematically rigorous) Uncertainty principle 0 0 ( )exp/ , p xip x 0 |( )| 1 p xx 0 0pandp Example 1: 0 0 xandx Example 2: 0 0 ( )() x xxx 00 0 11 ( )( )exp()exp() 22 xx ix pipx pxdx 0 1 |( )| 2 x pp Uncertainty principle 2 E Life time ( )E E E eigenstate Un
8、certainty principle: tunneling Therefore the kinetic energy of the particle must be greater than Vm-E Tunneling is a pure quantum effect due to uncertainty principle. x0X0+b Vm 2 ( ) 2 p EV x m 2 () m bb t v EEV m 2 E t The smaller the , the larger the Tunneling takes place in the moment that tE m E
9、EV 0VTE Whats wrong ? We have assumed that, at each instant we know both the kinetic and potential energy separately, i.e., we can assign values to x and p simultaneously. This is a classical picture. 242 2 2242 ()()0 4162 m EVE mbm bmb 2 2 8 m mb VE 22 22 2 1 24)8 ( ( ) 2 m mxm E b p V m 2 p x Unce
10、rtainty principle: tunneling Tunneling of an electron with energy E=2.979 a.u. through a single barrier of height Vm=5 a.u., and width b=1 a.u. The wave function plot (real and imaginary parts) corresponds to the following values of the coefficients: A1=1 (as a reference), B1=0.179-0.949i, A2=1.166-
11、0.973i, B2=0.013+0.024i, A3=-0.163-0.200i 22 111 ( ) ixmEixmE xAeBe 2()2() 222 ( ) ixm E Vm m ixm E V xA eB e 2 33 ( ) ixmE xA e 2 2 2 0 d k dx 2 2 2mT k Plane wave: ( ) ikxikx xAeBe Uncertainty principle: tunneling Tunneling of an electron with energy E=2.979 a.u. through two barriers of height Vm=
12、5 a.u., and width b=1 a.u, the barrier separation is L=. This is a resonance case. The real part of the wave function (a) oscillates before the 1st barrier with amplitude 1, increases by a factor of ca 3.5 within the 1st barrier, between the barriers the function makes slightly more than about one p
13、eriod, decays in the second barrier and goes out of the barrier region with an amplitude representing about 100% of the starting amplitude. A similar picture follows from the imaginary part of the wave function (b). Seminal papers on QM L. de Broglie, Comptes Rendus 177, 507 (1923); Nature 112, 540
14、(1923) W. Heisenberg, Zeit. Physik 33, 879 (1925) E. Schrdinger, Ann. der Physik 79, 36 (1929); 79, 489 (1926); 80, 437 (1926); 81, 109 (1926); 79, 734 (1926) P. A. M. Dirac, Proc. Roy. Soc. (London) A144, 243 (1927); A144, 710 (1927). Nobel laureates LaureateTimeWork M. Planck1918Quantum of action
15、A. Einstein1921Photoelectric effect N. Bohr1922Atomic structure and radiation L. de Broglie1929Matter wave W. Heisenberg1932Matrix mechanics E. Schrdinger1933Wave mechanics P. A. M. Dirac1933Relativistic wave mechanics W. Pauli1945Exclusion principle M. Born1954Statistical interpretation of wave fun
16、ction Everlasting disgrace of the Nobel Prize Committee Spin (1925, Uhlenbeck Newton, ) 原理原理 方法方法 技术技术 优化优化 运行运行 “现代科技的重大突破越来越依靠科研仪器的进步,现代科技的重大突破越来越依靠科研仪器的进步, 实际上,实际上,真正有望做到世界领先水平的实验科学研究真正有望做到世界领先水平的实验科学研究 工作,必须依靠有自己特点的先进科研仪器工作,必须依靠有自己特点的先进科研仪器,特别需,特别需 要那些运用到许多要那些运用到许多新原理、新方法和新技术新原理、新方法和新技术的的自主研自主研
17、发仪器发仪器。在国际较量面前,不能寄希望于用别人开发。在国际较量面前,不能寄希望于用别人开发 的仪器设备的仪器设备开拓新疆土开拓新疆土”。 中国科学报中国科学报2012-03-16 陈宜瑜,国家自然科学基金委主任陈宜瑜,国家自然科学基金委主任 基础理论是理论化学的生命力基础理论是理论化学的生命力 (living-force) 计算应用是理论化学的驱动力计算应用是理论化学的驱动力 (driving-force) (国家重大科研仪器设备研制专项一期资助经费即达(国家重大科研仪器设备研制专项一期资助经费即达5.7亿亿元人民币)元人民币) 理论化学的理论化学的新理论新理论、新方法新方法、新算法新算法、
18、新软件新软件、新应用新应用 General remarks “Computer experiment” lDeciphering the code. The language of computational chemistry is littered with acronyms. What do these abbreviations stand for in terms of underlying assumptions and approximations ? lTechnical problems. How does one actually run the program and w
19、hat does one look for in the output ? lQuality assessment. How good is the number that has been calculated ? lComputers do not solve problems, people do ! The real strength of computational chemistry is to gain insights and rationalizations of a large class of molecules based on the calculated data.
20、 l“If five different computational methods give five widely different results, what has computational chemistry contributed ? You just pick up the number closest to experiments and claim that you can reproduce experimental data accurately.” “Computer experiment” Operators and their finite matrix rep
21、resentations linear algebra, differential and integral calculus Reference: A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, Inc., New York, 1996 Chapter 1 The Hamiltonian 2 2 1 ( )( ) 22 p EmvV rV r m pi 2 2 ( ) 2 HV r m
22、HE | | H E | | H | | eff H We will introduce suitable mathematical machinery to solve the equation Key mathematical points Basis, delta function Operators and matrix representation The representations between different bases are mutually related Properties of determinants The variational method is t
23、he most powerful mathematical tool in QM Mathematical machinery Vectors Matrices Orthogonal functions Operators The variation method Notations: linear algebra, differential and integral calculus 1.1 Linear algebra jj aae aaeea i ii 1 Thus, 3 1i ijij aOb 3 1j jiji aOb In another form, 3 2 1 333231 23
24、2221 131211 3 2 1 a a a OOO OOO OOO b b b 3 2 1 b b b b, 3 2 1 a a a a Supplementary Material The matrix representation of a rotational operator x y 12 1 2 0 i j -1-2 (x,y) (x,y) a b jizC )( 4 ijzC )( 4 kkzC )( 4 100 001 010 )( 4 kjikjizC jiij eOeO 0 1111 eOeO 1)( 2112 iieOeO Then, the matrix repres
25、entation of zC4 is 100 001 010 )( 4 zC z y x z y x 100 001 010 1.1.2 Matrices (1) The matrix multiplication rule If A is an NM matrix, B is an MP matrix, then C=AB is an NP matrix with PjNiBAC kj M k ikij , 1;, 1 1 (2) The adjoint of an matrix A, denoted by A +, is an MN matrix with elements * jiij
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