1、1量子化学量子化学Modern Quantum ChemistryModern Quantum Chemistry2Modern Quantum ChemistryOpen a Door to Molecular Science3Quantum ChemistryvWhat is Quantum Chemistry?/Quantum Chemistry applies quantum mechanics to solve problems in chemistry.4The Power of Quantum Chemistry+To calculate and predict various
2、molecular properties,such as geometry conformation,dipole moments,barriers to internal rotation,NMR,frequencies and intensities in spectra.+To predict properties of transition states and intermediates in chemical reactions and to investigate the mechanisms of chemical reactions.+To understand interm
3、olecular forces and the behavior of molecules in solutions and solids.+To calculate thermodynamic properties(e.g.,entropy and heat capacity).5Chapter 1 Mathematical Review6 Chapter 1 Mathematical Review It is the aim of this chapter to equip you with the necessary mathematical machinery.vVectorsvMat
4、ricesvOrthogonal functionsvOperatorsvThe variation method78orthogonal otherwise 0normalized ji if 1jiijjiee ie are orthonormal basis vectors.jjaaeaaeeaiii1Unit dyadic:iiiee1(completeness relation)9(3)The operator1.Definition:An operator O is an entity which when acting on a vectora converts it into
5、another vector bbaOor transfers a given function into another function)()(xfxfD,dxdD Linear operators:numbers)any arey and(x )(bOyaOxbyaxO31jjijiOeeOijjieOeO (the component of the vector ieOalong je)333231232221131211321321)()(OOOOOOOOOeeeeeeO10333231232221131211OOOOOOOOOOOis the matrix representati
6、on of the operator O in the basis ie.2.How the operator O acts on arbitrary vector?baO 313131331 jjjjjiijiiijijijiiebeaOaOeaeOaO11 Thus,31iijijaOb31jjijiaObIn another form,321333231232221131211321aaaOOOOOOOOObbb321bbbb,321aaaa12Supplementary MaterialThe matrix representation of a rotational operator
7、xy12120ij-1-2(x,y)(x,y)abjizC)(4ijzC)(4kkzC)(4 100001010)(4kjikjizCjiijeOeO01111eOeO1)(2112iieOeO13Then,the matrix representation of zC4 is100001010)(4zCzyxzyx1000010101415IfABBA,or 0,ABBABA,then A and B commute.Commutator:BAABBA,ABBABA,Anti-commutator:BAABBA,ABBABA,1.1.2 Matrices(1)The matrix multi
8、plication rule If A is an NM matrix,B is an MP matrix,then C=AB is an NP matrix with PjNiBACkjMkikij,1;,1116(2)The adjoint of an matrix A,denoted by A+,is an MN matrix withelements*jiijAAIf the elements of A are real,then A+is called the transpose of AABABProof:ijkkjikkjkkikkijkjiijABABABBACAB *17(3
9、)Some important definitions and properties of square matrices,1.Diagonal matrixijiiijAA2.The trace of the matrix AiiiAAtr3.The unit matrixAAIIA ijijI184.The inverse of the matrix A,denoted by A-1IAAAA115.A unitary matrix A AA1A real unitary matrix is called orthogonal.6.A Hermitian matrix is self-ad
10、joint,i.e.,AA jiijAA A real Hermitian matrix is called symmetric.191.1.3 DeterminantsFor a square matrix,ANN !122111111)1(detNiNNipNNNNAAAPAAAAAAiiP:a permutation operator that permutes the column indices N,3,2,1.ip:the number of transpositions required to restore a given permutationNiiii,321 to nat
11、ural order N,3,2,1.It is important only whether ip is an even or odd number.A permutation of the numbers N,3,2,1 is simply a way of orderingthese numbers,and there are!N distinct permutations of N numbers.Each permutation can be expressed as the product of some transpositions.20There are 3!=6 permut
12、ation operators in S3:Pipi321321 e 0312321211123321311231321321213321)31(21 2 1323212131 2Supplementary MaterialsS3 permutation group21Some important properties:1.If each element in a row(or column)is zero,0A.2.If iiiiijAA,then NNiiiAAAAA22113.A single interchange of any two rows(or columns)of adete
13、rminant changes its sign.Proof:AAAApAAAppAAApNiNNipNiNNipNiNNipiii)1()1()1()1()1(!12211!1221121!12112 In the above formulae,we use the relations21pppii21pppiiiipp)1()1()1(224.AA5.BAAB 6.If any two rows(columns)are equal,0A7.11 AA8.If IAA,then 1AA.239.If OUU and IUUUU,Then,O.10.MkNNNkkNNMkNkkMkNkkABC
14、AAABCAAABCAA12112222211111211MkNNNkNNNkNkkABAAABAAABAAC121222221111211241.1.4 N-Dimensional complex vector spaces1.Ket vectorsNii,2,1,(a complete ket basis vectors)Niiaia1Naaaa21is the matrix representation of the abstract vector a in the basis i.252.Bra vectorsiaaii*i:a complete bra basis*2*1Naaaai
15、s the matrix representation of the bra vector a in the basis i.3.The scalar product between a bra a and a ket bNiiiNbabbbaaababa1*21*3*2*1211*|NiiNiiiaaaaa264.The orthonormality of the basis ijNiiijibabjiaba1*(by definition),thus,ijji5.The completeness of the basisiii1 Proof:By definition,we have ij
16、iaaijaj ijiajiaja*jajaaj iiiaiiaia iiiiiaiaa*so that,iii1276.The act of an operator O on a ket vectorbaOFirst,we are concerned with?iOThis is simple,because jjijOjiOjjiOiO1Since Niiaia1,one obtainsjjjiijiNiijbjaOiOaaO1from which iijijaOb or bOa 28Similarly,the matrix representation of the operator B
17、AC is jBAijBAiCij|1|jBkkAikkjBkkAi kjkikBA OrABC 297.The act of an adjoint of the operator O on a bra vectorIf baO,thenProof:bOaFrom the definition,*aOccOaandbaOIt is obvious thatbcaOccbbccOa*Thus,bOa308.If an operator is hermitian,then OOProof:By definition,*jiijOOHence,jOiiOjjOi*so that OO311.1.5
18、Change of basisSuppose we have two complete orthonormal bases i and,ijji,1iii ,11.The transformation matrix between two bases is unitary.Define ii,theniiiiii1 iiiiii*1 From ijjiijjiji One can find1In an analogous way,from,one can show that1 Hence,is unitary.322.The relationship between the matrix re
19、presentations in two differentbases.Suppose O is the matrix representation of O in the basis i and the matrix representation of O in the basis.OO11OOijjjOii jijijiOThus,OOiiithen,If331.1.6 The eigenvalue problemIf aaOa(:a constant),then is an eigenvector of the operator O with an eigenvalue.Specific
20、ally,if O is a hermitian operator,its eigenvectors and eigenvalueshave the following properties:1.The eigenvalues of a hermitian operator are real.Proof:aOa*OO By definition,OOThus,*aa342.The eigenvectors of a hermitian operator are orthogonal.Proof:Suppose O Othen*OOOO Therefore,0)(1)If,then 0.(2)I
21、f,two eigenvectors(they are said to be degenerate)canalways be chosen to be orthogonal(see supplementary materials).35Supplementary MaterialsSchmidt orthogonalizationProof:Assume that 0.111,0.122,021 S,and11O22O It is straightforward that)arbitraryare,(212121yxyxOyOxyxO36Schmidt orthogonalization co
22、nstructs orthogonal eigenvectors as follows:1Ized)(unnormali21CIIFrom the orthogonal condition 0III,one has012111CSCIIIwhich givesSC1After normalization,we have11111222SSCCSCSIIII1IIIIIIII 211112SS37383.Diagonalize the hermitian matrix OFrom the eigenvalue equation O,we have o Thus,in the basis form
23、ed by its eigenvectors,NO0021In general,a hermitian operator,say O,is not diagonal in an arbitrary basisNii,2,1,.39Solving an eigenvalue equation is identical to diagonalizing the hermitianmatrix O.The main idea of diagonalizing a matrix is to find a new basisN,2,1,in which the matrix representation
24、 of O,isdiagonal,i.e.,iiiNo002140The secular determinant method:From the above,NO0021Thus,kjkikijijOFurther,jijkjjkikkjkikkjkikijOOLet jiijC,jNjjjCCCC21,thenjjjCOC Nj,2,1COC01CO41A nontrivial solution(0C)exists only when0|1|OA secular determinant is a polynomial of degree N in the unknown,which has
25、N roots,j,Nj,2,1.Substituting each j into jjjCOC,one can find the correspondingeigenvectors jC.42Supplementary MaterialsThe eigenvalues and eigenvectors of the 22 symmetric matrix22211211OOOOO 2112OO(1)The secular determinant method022211211OOOOFrom this quadratic equation,we obtain21122112222111421
26、OOOOOO21122112222112421OOOOOO43Two eigenvectors can be deduced byjjjjjCCCCOOOO212122211211and12221jjCC(2)The Jacobi methodLet cossinsincos (which is a general 22 orthogonal matrix)cossinsincoscossinsincos22121211OOOOO2sincossin2cos2sin212cos2sin212sinsincos1222221112221112221112222211OOOOOOOOOOOO210
27、044Thus,02cos2sin21122211OOO22111222tanOOO22111202arctan21OOO0120222021112sinsincosOOO0120222021122sincossinOOOTwo eigenvectors are simply001sincosC002cossinC451.1.8 Functions of matrices1.DefinitionsLike xfx,one can define various functions for matrices AfA(1)The square root of a matrix A,denoted b
28、y 21A,AAA2121(2)The exponential of a matrix are defined as the Taylor series of thefunction,i.e.,32!31!21!111)exp(AAAA(3)In general,0)(nnnACAf462.How to calculate functions of matrices?(1)If ijiijaA,then nNnnnaaaA0021so that 02100nnnNnnnnnnnnnaCaCaCACAf Nafafaf0021For example,2/12/122/112/100NaaaA47
29、(2)If A is not diagonal,we must find a unitary transformation thatdiagonalizes it,i.e.,NaaaaA0021Since the reverse transformation yieldsaA22aaaAor in generalnnaAwe obtain nNnnnnnafafafaCACAf0021For example,AaAAaA2/12/12/12/1481.The role played by a set of orthogonal functions is similar to the rolep
30、layed by a set of basis vectors in N-dimensional complex vectorspace.Given an infinite set of functions ,3,2,1,ixi which satisfy theorthonormality condition21)()(xxijjixxdxAn arbitrary function can be expanded in terms of,3,2,1,ixi:49By introducing Dirac delta function xx,iiixxxx)()()(Hence,)()()(xa
31、xxdxxaBecause xx is real,)()(xxxxAs long as the integration interval includes 0 x,we have)()()()()0(xxdxaxxadxaLet 1)(xa,then )(1xdxOne of forms to express the delta function is)()(lim0 xxwhereotherwise 0 x-21)(x502.The theory of complete orthonormal functions can be regarded as ageneralization of o
32、rdinary linear algebra.ixi ixi*axa axa*Orthonormal functionsLinear algebra baxbxdxa*jjaajxaxdx*iiiiaiiaaxxa baOxbxaO)(),()()(xaxxOdxxaOxbOabaObjjiji OxxO 11*xxdx OxOxdx*bOaxbOxdxa*51Dirac notation allows us to manipulate vectors and functions,as well as theoperators acting on them,in a formally iden
33、tical way.For a general hermitian operator,its exact solutions are usually unknown.We need to develop approximate methods.H ,1,0Suppose its exact eigenvalues can be arranged in the order,21oObviously,its exact eigenvectors,.2,1,0,form a completebasis set,i.e.,521.3.1 The variation principleGiven a n
34、ormalized approximate wave function,then0 H1Proof:Since C HH Since 0 for all,one obtains000HUsing the normalization condition 1,oHwhich is the required result.53In an analogous way,it can be shown thatexacteapproximatE,2,11.3.2 The linear variational problemFor an arbitrary wave function,suppose it
35、can be expanded asiNiiC 1where i is a fixed set of N basis functions.Assume that the basis functions are real and orthonormal,i.e.,ijijjiThen,12iijijijiCCCijjijijjiijiHCCCHCH541.The linear variational problem is to minimize H subject to theconstraint 1,i.e.,11,2,102iikCNkHCThe Lagranges method of un
36、determined multipliers can be used to solve theproblem:121HCCCN，iiijjijiCHCC12From 0kC,NNk,1,2,1,we have 02kikiikjjjkCHCHCCNotice:kkkikKiikkjkjjkijjkijiijjijiHCHCCHCCHCCHCC2)(550kiikiCCHcHc56CCHCC572.An alternative way to derive the above resultAs we know,exact eigenfunctions satisfyH.Assume thiseig
37、envalue equation also holds for approximate eigenfunctionsNjjjC1,i.e.,EHThus,jjjjjjCEHC1Multiplying by i on the left yieldsijijjjijjECCEHC ijjijECCHDefine NCCCc21,the above equation becomes EcHc Thus the linear variational method is equivalent to solving the eigenvalueequation,H,in a finite subspace spanned byNii,2,1,.
