Part III Symmetry and BondingChapter 3DirectProducts第三章(表示的）直积Prof.Dr.XinLu（吕鑫）Email:xinlu@xmu.edu.cnhttp://pcoss.xmu.edu.cn/xlv/index.htmlhttp://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
Part III Symmetry and Bonding Chapter 3 Direct Products 第三章 （表示的）直积 Prof. Dr. Xin Lu （吕鑫） Email: xinlu@xmu.edu.cn http://pcoss.xmu.edu.cn/xlv/index.html http://pcoss.xmu.edu.cn/xlv/courses/theochem/index.html
3. Direct products.In this chapter we will learn how to find the symmetry of a product of two or more functions.Thisis extraordinarily important!Sy=[viwjdtβ=J s,HspdtRecallthoseintegralsweusedbefore.3.1 Introduction.From the Ca.charactertable, we know that the function xtransforms likeB,whereas thefunction y transforms like B,.Then how does the function xy transform?9J2EgxzC2C2v: This is already given in the tableThefunctionxytransformslikeA,11A111x2.Z1A21-1-1Rzxy.How canweactuallyworkthisout?B11-11-1RyxXz1B21-1-1Rxyyz
3. Direct products • In this chapter we will learn how to find the symmetry of a product of two or more functions. This is extraordinarily important! Recall those integrals we used before: • From the C2v character table, we know that the function x transforms like B1 whereas the function y transforms like B2 . Then how does the function xy transform? 3.1 Introduction • This is already given in the table. The function xy transforms like A2 . • How can we actually work this out? 𝑺𝒊𝒋 = ψ𝒊 ∗ψ𝒋𝒅𝝉 𝜷 = sa𝑯 sbdτ
3.1 Direct products introductionUse the function xy as a basis to form the corresponding representation of C2w which will just be asetofnumbers,i.e.,these numbers arethe characterseffectofoyzeffect of C2zeffectofoxzstartVe2xEx = (+1)x,C3x = (-1)x,ox=(+1)x,ox=(-1)x，(1,-1,1,-1) B,yEy =(+1)y;C3y = (-1)y;oy =(-1)y,(1,-1,-1,1) B2ay= (+l)y,xyE(xy) = (+1)xy;C,xy =(+1)xy;oxzxy =(-1)xy,axy =(-1)xy, (1,1,-1,-1) A,oxoJEC2C2v1x2;y2;z2A1111Z> xy transforms like A,11-1-1R.A2xyBi-1R,11-1xxzRx1B21-1-1yyz
3.1 Direct products introduction • Use the function xy as a basis to form the corresponding representation of C2v, which will just be a set of numbers, i.e., these numbers are the characters. xy Ex = (+1)x, Ey =(+1)y; E(xy) = (+1)xy; C2 zx = (-1)x, C2 zy = (-1)y; C2 zxy = (+1)xy; xzx = (+1)x , xzy = (-1)y, xzxy =(-1)xy, yzx = (-1)x , yzy = (+1)y, yzxy = (-1)xy, y x (1,-1,1,-1) B1 (1,-1,-1,1) B2 (1,1,-1,-1) A2 xy transforms like A2
: The characters for xy are simply found by multiplying together the characters for the IR B, whichis how x transforms, and for the IR B,, which is how y transforms, operation by operation:(1, -1, 1, -1)@(1, -1, -1, 1) = (1 × 1, -1 × -1, 1 × -1, -1 × 1)=(1,1, -1, -1)Bi (x)B2 (y)B1@B2=A2Thiskind ofmultiplicationis calledthedirectproduct:B,B,=A,.Totakeanotherexample,ifwewantedtoknowhowxztransforms:(1, -1, 1, -1)@(1, 1, 1, 1) = (1 × 1, -1 × 1, 1 × 1, -1 × 1) = (1, -1, 1, -1)BIAI=BIBi (x)Al (z)zoyzEC2C2vThusxztransforms likeB,1111x2.1A1zA21-1Rz1-1xy1-11Bi-1RyxxzB2Rx1-1-1yyz
• The characters for xy are simply found by multiplying together the characters for the IR B1 , which is how x transforms, and for the IR B2 , which is how y transforms, operation by operation: This kind of multiplication is called the direct product: B1⊗ B2= A2 . • To take another example, if we wanted to know how xz transforms: Thus xz transforms like B1
3.2 Direct product of one-dimensional irreducible representationsOne-dimensional IRs arethose with character1 under the operationEand always denoted by the labels A and B全对称不可约表示In any group there is always the totally symmetric IR with all of the characters being +1For the ith one-dimensional IR, , of a group, the following properties apply:1) The direct product of this IR with the totallysymmetric IR, Itot. sym, gives this IR,oxzyzEC2C2v@rtot sym=x2;y2;221111AZRz11-1A2-1xy2)Thedirectproductofaone-dimensional IR1B1-11-1R,xXzwithitselfgivesthetotallysymmetricIR1B2-1-11Rxyyz[i?(i)=rtot. sym
3.2 Direct product of one-dimensional irreducible representations One-dimensional IRs are those with character 1 under the operation E, 1) The direct product of this IR with the totally symmetric IR, Γtot. sym. , gives this IR, 2) The direct product of a one-dimensional IR with itself gives the totally symmetric IR For the ith one-dimensional IR, Γ(i) , of a group, the following properties apply: Γ(i)⊗ Γtot. sym.= Γ(i) Γ(i)⊗ Γ(i)= Γtot. sym. and always denoted by the labels A and B. In any group there is always the totally symmetric IR with all of the characters being +1. 全对称不可约表示