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
26、AA If the elements of A are real, then A+ is called the transpose of A ABAB Proof: ij k kjik k jkki k kijkjiij AB AB ABBACAB * It is important only whether i p is an even or odd number. Each permutation can be expressed as the product of some transpositions. Supplementary Materials S3 permutation gr
27、oup 1. If each element in a row (or column) is zero, 0A. 2. If iiiiij AA, then NN i ii AAAAA 2211 3. A single interchange of any two rows (or columns) of a determinant changes its sign. Proof: A AAAp AAAppAAAp N i NNi p N i NNi p N i NNi p i ii ) 1( ) 1() 1( ) 1() 1( ! 1 2211 ! 1 221121 ! 1 2112 In
28、the above formulae, we use the relations 21 ppp ii 21 ppp ii ii pp ) 1() 1() 1( 9. If OU U and IUUUU , Then, O. 10. M k NNNkkNN M k Nkk M k Nkk ABCAA ABCAA ABCAA 1 21 1 222221 1 111211 M k NNNkNN Nk Nk k ABAA ABAA ABAA C 1 21 222221 111211 (a determinant is invariant w.r.t. the unitary transformatio
29、n of the matrix) 1.1.4 N-Dimensional complex vector spaces 1. Ket vectors Nii, 2 , 1, (a complete ket basis vectors) N i i aia 1 N a a a a 2 1 is the matrix representation of the abstract vector a in the basis i. 2. Bra vectors iaa i i * i: a complete bra basis * 2 * 1N aaaa is the matrix representa
30、tion of the bra vector a in the basis i. 3. The scalar product between a bra a and a ket b N i ii N ba b b b aaababa 1 * 2 1 * 3 * 2 * 1 2 11 * | N i i N i ii aaaaa 4. The orthonormality of the basis ij N i iiji babjiaba 1 * (by definition), thus, ij ji 5. The completeness of the basis Proof: By def
31、inition, we have i ji aaijaj i ji ajiaja * * jajaaj ii i aiiaia ii i iiaiaa * so that, ii i 1 baO First, we are concerned with ? iO Or ABC From the definition, * aOccOa and baO It is obvious that bcaOc cbbccOa * Thus, bOa 8. If an operator is hermitian, then OO Proof: By definition, * jiij OO Hence,
32、 jOiiOjjOi * so that OO All the above shows that an operator is equivalent to its matrix representation in a complete basis. The spectral resolution is | ij ij OiOj Summary: Discretization of an operator If we have a complete basis | ij i jS 1 | 1 ij ij iSj 11 (|) (|) ijij ijkl OiSj OkSl 11 | ijjkkl
33、 ijkl iS O Sl | ij ij iOj | ijij Si j if| 1 i ii 1.1.5 Change of basis Suppose we have two complete orthonormal bases i and , ij ji, 1 i ii , 1 2. The relationship between the matrix representations in two different bases. Thus, O Oi i i then,If 2. The eigenvectors of a hermitian operator are orthog
34、onal. Proof: Suppose O O then * O OO O Therefore, 0)( (1) If , then 0. (2) If , two eigenvectors (they are said to be degenerate) can always be chosen to be orthogonal (see supplementary materials). Supplementary Materials Schmidt orthogonalization Proof: Assume that 0 . 111, 0 . 122, 021 S, and 11
35、O 22 O It is straightforward that )arbitraryare,( 212 1 21 yx yxOyOxyxO After normalization, we have 11111 222 SSCCSCSIIII 1 II IIII II 21 1 1 1 2 S S 3. Diagonalize the hermitian matrix O From the eigenvalue equation O , we have o i i i N o 0 0 2 1 The secular determinant method: From the above, N
36、O 0 0 2 1 Thus, kj k ikijij O Further, jijkjj k ikkj k ikkj k ikij OO Let j iij C, j N j j j C C C C 2 1 , then j j j COC Nj, 2 , 1 COC 01CO Supplementary Materials (1) The secular determinant method 0 2221 1211 OO OO From this quadratic equation, we obtain Two eigenvectors can be deduced by j j j j
37、 j C C C C OO OO 2 1 2 1 2221 1211 and 1 2 2 2 1 jj CC (2) The Jacobi method Let cossin sincos (which is a general 22 orthogonal matrix ) 1112 1222 cossincossin sincossincos OO O OO 2sincossin2cos2sin 2 1 2cos2sin 2 1 2sinsincos 12 2 22 2 11122211 12221112 2 22 2 11 OOOOOO OOOOOO 2 1 0 0 Thus, 02cos
38、2sin 2 1 122211 OOO 2211 12 2 2tan OO O 2211 12 0 2 arctan 2 1 OO O 0120 2 220 2 111 2sinsincosOOO 0120 2 220 2 112 2sincossinOOO Two eigenvectors are simply 0 0 1 sin cos C 0 0 2 cos sin C 1. Definitions 0 )( n n nA CAf 2. How to calculate functions of matrices? (1) If ijiij aA, then n N n n n a a
39、a A 0 0 2 1 so that 0 2 1 0 0 n n n Nn n n n n n n n n aC aC aC ACAf N af af af 0 0 2 1 For example, 2/1 2/1 2 2/1 1 2/1 0 0 N a a a A (2) If A is not diagonal, we must find a unitary transformation that diagonalizes it, i.e., N a a a aA 0 0 2 1 Since the reverse transformation yields aA 22 aaaA or
40、in general nn aA we obtain 1 2 0 0 ( ) nn nn nn N f A f a f a C AC f a f aa For example, AaAAaA 2/12/12/12/1 Given that ,1,2,U AUaorAca CN 1 ( )( 1)GA the matrix element of reads Or in Dirac notation 1 | ( ) N G a ( 1)A xb We can use these relations to solve the following set of inhomogeneous linear
41、 system of equations: 1 ( 1)( )xAbGb * 1 ( )( ( ) NN ijj iijj jj U U b xGb a * 11 ( ( ) NN ijij ij U Uc c G aa 1 ( )( 1)U GUa 1. The role played by a set of orthogonal functions is similar to the role played by a set of basis vectors in N-dimensional complex vector space. Given an infinite set of fu
42、nctions , 3, 2, 1, ix i which satisfy the orthonormality condition 2 1 )()( x x ijji xxdx An arbitrary function can be expanded in terms of , 3, 2, 1, ix i : By introducing Dirac delta function xx, i ii xxxx) () ()( Hence, ) () ()(xaxxdxxa Because xx is real, )() (xxxx As long as the integration int
43、erval includes 0 x, we have )()() () ()0(xxdxaxxadxa Let 1)(xa, then )(1xdx One of forms to express the delta function is )()( lim 0 xx where otherwise 0 x- 2 1 )( x 2 1 2 0 x 2. The theory of complete orthonormal functions can be regarded as a generalization of ordinary linear algebra. ix i ix i *
44、axa axa * Orthonormal functions Linear algebra Dirac notation allows us to manipulate vectors and functions, as well as the operators acting on them, in a formally identical way. H , 1, 0 Suppose its exact eigenvalues can be arranged in the order, 21o Obviously, its exact eigenvectors ,.2 , 1 , 0, f
45、orm a complete basis set, i.e., 1.3.1 The variation principle Given a normalized approximate wave function , then 0 H 1 Proof : Since C HH Since 0 for all , one obtains 0 00 H Using the normalization condition 1 , o H which is the required result. In an analogous way, it can be shown that exacteappr
46、oximatE , 2, 1 1.3.2 The linear variational problem For an arbitrary wave function , suppose it can be expanded as i N i i C 1 where i is a fixed set of N basis functions. Assume that the basis functions are real and orthonormal, i.e., ijijji Then, 1 2 i ijij ij i CCC ijj ij ijji ij i HCCCHCH |1,|0,
47、 i i 1. The linear variational problem is to minimize H subject to the constraint 1 , i.e., 1 1, 2 , 10 2 i i k C NkH C The Lagranges method of undetermined multipliers can be used to solve the problem: 1 21 HCCC N， ， i iijj ij i CHCC1 2 Choose to make0 N C 0 ki i ki CCH cHc CCHCC 2. An alternative
48、way to derive the above result As we know, exact eigenfunctions satisfyH . Assume this eigenvalue equation also holds for approximate eigenfunctions N j jj C 1 , i.e., EH Define N C C C c 2 1 , the above equation becomes EcHc Summary: mathematical machinery Basis, delta function Operators and matrix
49、 representation The representations between different bases are mutually related Properties of determinants The variational method is the most powerful mathematical tool in QM Many-Electron Wave Function and operators Reference: A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to A
50、dvanced Electronic Structure Theory, Dover Publications, Inc., New York, 1996 Basic concepts, techniques, and notations of QC: The structure of the Hamiltonian, the form of the wave function, the matrix elements of the Hamiltonian between determinants, the Hartree-Fock method, etc. Outline 2.1 The E
