M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 33
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Another important point is that the character of amatrix is not changed by any similarity transformation. From this itfollows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representationfrom which they are obtained.
We have seen that for each symmetryoperation the matrices of the irreducible representations stand alongthe diagonal of the matrix of the reducible representation, and thecharacter is just the sum of the diagonal elements. When reducinga representation, the simplest way is to look for the combination ofthe irreducible representations of that group—that is, the sum of theircharacters in each class of the character table—that will produce thecharacters of the reducible representation.First, reduce the representation of the two N–H bond lengthchanges of HNNH:⌫12002The C2h character table shows that ⌫1 can be reduced to Ag + Bu :C2hEC2ihAgBgAuBu11111–11–111–1–11–1–11Ag + Bu2002It may be asked, of course, whether this is the only way of decomposing the ⌫1 representation. The answer is reassuring: The decomposition of any reducible representation is unique.
If we find a solutionjust by inspection of the character table, it will be the only one. Oftenthis is the fastest and simplest way to decompose a reducible representation.A more general and more complicated way is to use a reductionformula:4.8. Reducing a Representation207ai = (1/ h)χ (R) · χi (R)Rwhere ai is the number of times the ith irreducible representationappears in the reducible representation, h is the order of the group,R is an operation of the group, χ(R) is the character of R in the∗reducible representation and χ i (R) is the character of R in the ithirreducible representation.
The summation extends over all operationsof the group.The reduction formula can be simplified by grouping the equivalentoperations into classes,N · χ(R) Q · χi (R) Qai = (1/ h)Qwhere ai is the number of times the ith irreducible representationappears in the reducible representation, h is the order of the group,Q is a class of the group, N is the number of operations in class Q,R is an operation of the group, χ (R)Q is the character of an operationof class Q in the reducible representation, and χ i (R)Q is the characterof an operation of class Q in the ith irreducible representation.
Thesummation extends over all classes of the group.The reduction formula can only be applied to finite point groups.For the infinite point groups, D∞h and C∞h , the usual practice is toreduce the representations by inspection of the character table.For illustration, let us find the irreducible representations of the twoexamples used before. First, on the basis of the two N–H distancechanges of diimide (i.e., ⌫1 ):∗C2hEC2ihAgBgAuBu11111–11–111–1–11–1–11⌫12002Here and hereafter the short expression “character of R” stands for the character ofthe matrix corresponding to operation R, in accordance with our previous discussion.2084 Helpful Mathematical ToolsThe order of the group is 4.
The number of times the irreduciblerepresentation Ag appears in the reducible representation isa Ag = (1/4)[1 · 2 · 1 + 1 · 0 · 1 + 1 · 0 · 1 + 1 · 2 · 1]= (1/4)(2 + 0 + 0 + 2) = 4/4 = 1In the same way we can deduce the number of times the other irreducible representations appear in ⌫1 :a Bg = (1/4)[1 · 2 · 1 + 1 · 0 · (−1) + 1 · 0 · 1 + 1 · 2 · (−1)] = 0a Au = (1/4)[1 · 2 · 1 + 1 · 0 · 1 + 1 · 0 · (−1) + 1 · 2 · (−1)] = 0a Bu = (1/4)[1 · 2 · 1 + 1 · 0 · (−1) + 1 · 0 · (−1) + 1 · 2 · 1] = 1That is, ⌫1 = A g + Bu , and the result is the same as before.With the 12-dimensional reducible representation of the Cartesiandisplacement vectors of HNNH, the inspection method probably doesnot work.
However, the reduction formula can be used. The reduciblerepresentation is:⌫212004and with applying the reduction formula, we obtain:a Ag = (1/4)[1 · 12 · 1 + 1 · 0 · 1 + 1 · 0 · 1 + 1 · 4 · 1] = 4a Bg = (1/4)[1 · 12 · 1 + 1 · 0 · (−1) + 1 · 0 · 1 + 1 · 4 · (−1)] = 2a Au = (1/4)[1 · 12 · 1 + 1 · 0 · 1 + 1 · 0 · (−1) + 1 · 4 · (−1)] = 2a Bu = (1/4)[1 · 12 · 1 + 1 · 0 · (−1) + 1 · 0 · (−1) + 1 · 4 · 1] = 4Thus,⌫2 = 4A g + 2Bg + 2Au + 4Bu4.9. AuxiliariesA few additional things need to be mentioned before embarkingon chemical applications of group theoretical methods. For detaileddescriptions and proofs we refer to References [21–23].4.9. Auxiliaries2094.9.1. Direct ProductWave functions form bases for representations of the point group ofthe molecule [24]. Suppose that fi and fj are such functions; then thenew set of functions, fi fj , called the direct product of fi and fj , is alsobasis for a representation of the group.
The characters of the directproduct can be determined by the following rule: The characters ofthe representation of a direct product are equal to the products of thecharacters of the representations of the original functions. The directproduct of two irreducible representations will be a new representationwhich is either an irreducible representation itself or can be reducedinto irreducible representations. Tables 4-8 and 4-9 show some examples for direct products with the C2v and C3v point groups, respectively.Table 4-8. Character Table and Some Direct Products for the C2v Point GroupC2EC2A1A2B1B2111111−1−11−11−11−1−11A1 ·A2A2 ·B1B1 ·B21111−11−1−1−1−11−1= A2= B2= A2Table 4-9.
Character Table and Direct Products for the C3v Point GroupC3E2C33vA1A2E11211−11−10A2 ·A2A2 ·EE·E1241−11100= A1=E= A1 + A2 + E4.9.2. Integrals of Product FunctionsIntegrals of product functions often occur in the quantum mechanical description of molecular properties and it is helpful to knowtheir symmetry behavior. Why? The reason is that an integral whose2104 Helpful Mathematical Toolsintegrand is the product of two or more functions will vanish unlessthe integrand is invariant under all symmetry operations of the pointgroup.
There is only one irreducible representation whose charactersare 1 for each symmetry operation of the point group, and this is thetotally symmetric irreducible representation. Therefore, an integralwill be nonzero only if the integrand belongs to the totally symmetricirreducible representation of the molecular point group.The representation of a product function can be determined byforming the direct product of the original functions. The representation of a direct product will contain the totally symmetric representation only if the original functions whose product is formed belongto the same irreducible representation of the molecular point group.This follows directly from rules 2 and 3 in Section 4.5.These rules can be extended to integrals of products of more thantwo functions.
For a triple product the integral will be nonzero only ifthe representation of the product of any two functions is the same as,or contains, the representation of the third function. If the integral isf i · f j · f k dτthen the above condition is expressed by⌫ fi · ⌫ fk ⊂ ⌫ f jwhere ⌫ stands for the representation and ⊂ means “is or contains.”Very often, fj is a quantum-chemical operator, and then the expressionsare:ˆ f k dτf i op.or with other notation,ˆ fk f i |op.|and⌫ fi · ⌫ fk ⊂ ⌫op·ˆThis kind of condition appears in energy integrals and spectralselection rules, and in the discussion of chemical reactions.4.9.
Auxiliaries2114.9.3. Projection OperatorThe projection operator is one of the most useful concepts in theapplication of group theory to chemical problems [25, 26]. It is anoperator which takes the non-symmetry-adapted basis of a representation and projects it along new directions in such a way that it belongsto a specific irreducible representation of the group. The projectionoperator is represented by P̂ i in the following form:P̂ i = (1/ h)χi (R) · R̂Rwhere h is the order of the group, i is an irreducible representationof the group, R is an operation of the group, χi (R) is the character ofR in the ith irreducible representation, and R̂ means the applicationof the symmetry operation R to our basis component.
The summationextends over all operations of the group.Consider now the construction of the A1 symmetry group orbitalof the hydrogen s atomic orbitals in ammonia as an example of theapplication of the projection operator. (The various kinds of orbitalswill be discussed in detail in Chapter 6.) The projection operator forthe A1 irreducible representation in the C3v point group isP̂ A1 = (1/6)χ A1 (R) · R̂RApplying this operator to the s orbital of one of the hydrogens (H1)of ammonia, we obtainP̂ A1 s1 ≈ 1 · E · s1 + 1 · C3 · s1 + 1 · C32 · s1 + 1 · · s1+ 1 · · s1 + 1 · · s1= s1 + s2 + s3 + s1 + s2 + s3 ≈ s1 + s2 + s3This expression is an approximation since the numerical factor of1/6 was omitted.
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