《无机化学 Chemical Theory》课程教学资源（讲稿）Part II Molecular Spectroscopy Chapter 3 Direct Products 直积

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

3.3Direct product oftwo-dimensional irreducible representations. Two-dimensional IRs have character 2 under the identity operation, and are always denoted by alabel E.(e.g.,EIR in C3y)C3vE2C30V11x2 +y2;1AlProperty1fromtheprevioussectionstillz11Rz-1A2applies.Forexample,ifwetakethedirectE20-1(x,y)(Rx,R,)(xz, yz); (x2 - y2,2xy)product A,E we obtain E410EOE=E ④A,④A,(1, 1,1)@(2, -1, 0) = (1 × 2, 1 × -1, 1 × 0) = (2, -1, 0)(2, -1, 0) (2, -1,0) = (2 × 2, -1 × -1,0 ×0) = (4, 1, 0)EEE)AlE: Property 2 does not apply. If we compute E E, we findEE=E@A,A,? Modified version of property 2: The direct product of an IR with itself contains the totallysymmetricIR.This trend holdsfor higher-dimensional IRs

3.3 Direct product of two-dimensional irreducible representations • Two-dimensional IRs have character 2 under the identity operation, and are always denoted by a label E. （e.g., E IR in C3v) • Property 1 from the previous section still applies. For example, if we take the direct product A1⊗E we obtain E. • Property 2 does not apply. If we compute E ⊗ E , we find E ⊗ E = E  A1 A2 • Modified version of property 2: The direct product of an IR with itself contains the totally symmetric IR. This trend holds for higher-dimensional IRs. E ⊗ E 4 1 0 = E  A1 A2

3.4 Further points: How does xyz transforms in the group Ca? Consider the triple direct product:Eo-xzXC2vC2B1A2?A1B2?Al&=A2x2;y2;z2A11111zB1&B2ZxyZA211-1-1RxyBi1-11-1R,xxzThusxyztransforms as A,B2-11-11RxyyzThedirectproductiscommutativeanddistributivei.e. B,B,-B,B, and (B,B) @A,=B,(B,QA): Simple numbers (scalars) transform as the totally symmetric IR, as a number isunaffectedby any symmetry operation.Ex.13

3.4 Further points • How does xyz transforms in the group C2v? Consider the triple direct product: • The direct product is commutative and distributive. Thus xyz transforms as A2 . = A2 . • Simple numbers (scalars) transform as the totally symmetric IR, as a number is unaffected by any symmetry operation. i.e. B1⊗ B2= B2⊗ B1 and (B1⊗B2 ) ⊗ A1= B1⊗ (B2⊗A1 ). Ex.13

3.5 Summary: If two functions transform as the IRs (iand r), respectively, then their producttransforms as the direct product of the two IRs o): The direct product is found by multiplying the characters of the two IRs for eachsymmetry operation: (a, b, c, . . .) @(p, q, r, .. ) = (axp, b xq, cxr, . . .). The totally symmetric IR, Frot sym, has character +1 for all operations.. For any IR i: @rot. sym= [(i).For any one-dimensional IR:i @Ti)=Trot sym.: For any higher-dimensional IR the result of the product I(i @r(i) contains Ftot sym. Scalars (numbers) transform as the totally symmetric IR

3.5 Summary • If two functions transform as the IRs Γ(i)and Γ(j), respectively, then their product transforms as the direct product of the two IRs Γ(i)⊗ Γ(j) . • The direct product is found by multiplying the characters of the two IRs for each symmetry operation: (a, b, c, . . .) ⊗ (p, q, r, . . .) = (a×p, b×q, c×r, . . .) • The totally symmetric IR, Γtot. sym. , has character +1 for all operations. • For any IR Γ(i) : Γ(i)⊗ Γtot. sym.= Γ(i) . • For any one-dimensional IR: Γ(i)⊗ Γ(i)= Γtot. sym. • For any higher-dimensional IR the result of the product Γ(i)⊗Γ(i ) contains Γtot. sym. . • Scalars (numbers) transform as the totally symmetric IR