M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 30
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Working with such bigmatrices is awkward and time-consuming. Fortunately, they can besimplifed. We shall not go into the details of how this is done sinceonly the easiest and quickest methods utilizing matrix representationswill be used in the next chapters. We shall merely outline the procedure leading from the big unpleasant representations of symmetryoperations to simpler tools [12]. With the help of suitable similaritytransformations, matrices can be turned into so-called block-diagonalmatrices. A block-diagonal matrix has nonzero values only in squareblocks along the diagonal from the top left to the bottom right.
Themerits of block-diagonal matrices are best illustrated in their multiplication. Suppose, for example, that two 5 × 5 matrices are to bemultiplied, as follows:⎡2⎢1⎢⎢0⎢⎣00320000011000110⎤ ⎡01⎢20⎥⎥ ⎢⎢0⎥⎥ · ⎢00 ⎦ ⎣020210000021000220⎤ ⎡08⎢50⎥⎥ ⎢⎢0⎥⎥ = ⎢00⎦ ⎣010740000033000440⎤00⎥⎥0⎥⎥0⎦21884 Helpful Mathematical ToolsThe determination of the first row is already quite complicated:2·1+3·2+0·0+0·0+0·0=82·2+3·1+0·0+0·0+0·0=72·0+3·0+0·2+0·1+0·0=02·0+3·0+0·2+0·2+0·0=02·0+3·0+0·0+0·0+0·1=0Notice that the product of two equally block-diagonalized matrices—such as those two above—is another similarly block-diagonalizedmatrix. It is especially important that this resulting matrix can beobtained simply by multiplying the corresponding individual blocksof the original matrices.
Check this on the above example: 2 31 22·1+3·2 2·2+3·18 7·==1 22 11·1+2·2 1·2+2·15 1 1 12 21·2+1·1·=1 11 21·2+1·1 1·2+1·23 4=1·2+1·23 4[2] · [1] = [2]Generally, if two matrices A and B can be transformed by similaritytransformation into identically shaped block-diagonalized matrices,their product matrix C will also have the same block-diagonal form:⎤ ⎡⎤ ⎡⎤⎡A1B1C1⎣ A2 ⎦ · ⎣ B2 ⎦ = ⎣ C2 ⎦A3B3C3The multiplication will also be valid for the individual blocks:A1 · B1 = C1A2 · B2 = C2A3 · B3 = C3Since the blocks themselves will obey the same multiplication tablethat the big matrices do, each block will be a new representation for an4.4.
The Character of a Representation189operation of the group. Thus, if the above A and B matrices are representations for the respective symmetry operations v and v in the C2vpoint group, so will be the matrices A1 , A2 , and A3 and B1 , B2 , and B3 ,respectively. The C2v multiplication table (Table 4-1) shows thatv · v = C2and, accordingly, not only the big C matrix but also the small matricesC1 , C2 , and C3 , will be representations of the C2 operation. This waythe big matrices reduce into smaller ones which are more convenientto handle. Let us suppose that the above big matrices A, B, and Ctogether with the E matrix constitute a representation for the C2v pointgroup. This is called then a reducible representation of the group,indicating that it is possible to find a similarity transformation thatreduces all its matrices into new ones with smaller dimension. If thisis repeated until it is no longer possible to find a similarity transformation to reduce simultaneously all the matrices of a representationinto smaller ones, we call this representation irreducible.
Suppose nowthat in the example above the small matrices along the diagonals ofthe big ones cannot be reduced further by a similarity transformation. In this case each set of the small matrices will be an irreduciblerepresentation of the C2v point group. The set of A1 , B1 , C1 , and E1will be an irreducible representation, so will be the set of A2 , B2 , C2 ,and E2 , and yet another irreducible representation will be the set ofA3 , B3 , C3 , and E3 . Thus, the reducible representation was reduced tothree irreducible representations. Since the symmetry operations canbe applied to all kinds of bases for a molecule, there may be countlessnumbers of reducible representations.
The important thing is that allthese representations reduce into a small and finite number of irreducible representations for practically all point groups. These irreducible representations, often called symmetry species, are then usedin many areas of chemistry to describe symmetry properties.4.4. The Character of a RepresentationConsidering the sizes of the initial matrices, using irreducible representations is a great improvement. Fortunately, even further simplification is possible.
Instead of working with irreducible representations1904 Helpful Mathematical Toolswe can use simply their characters. The utility of this approach will∗be amply demonstrated later. The character of a matrix is the sum ofits diagonal elements. For the following matrix1203071112001–23 –4the character is:1 + 7 + 0 + (−4) = 4Since a representation—reducible or irreducible—is a set ofmatrices corresponding to all symmetry operations of a group, therepresentation can be described by the set of characters of all thesematrices. For the simple basis of ⌬r1 and ⌬r2 used before for theHNNH molecule in the C2h point group, the representation consistedof four 2 × 2 matrices:E=10010i=11h =0C2 =characters011010011+1=20+0=00+0=01+1=2Thus, the characters of this representation are2∗0In linear algebra this is usually called trace.024.5.
Character Tables and Properties of Irreducible Representations191We do not know yet, however, whether this representation isreducible or irreducible. To answer this question, first we have toknow the characters of the irreducible representations of the C2h pointgroup.4.5. Character Tables andProperties of Irreducible RepresentationsThe characters of irreducible representations are collected in so-calledcharacter tables.
We shall not discuss here how to find the characters of a given irreducible representation. The character tables arealways available in textbooks and handbooks, or on the Internet, andsome of them are also given in the subsequent chapters of this book.Table 4-3 shows the character table for the C2h point group. The toprow contains the complete set of symmetry operations of this group.The left column shows, for the time being, some temporary names.⌫ is the generally used label for the representations.
The main bodyof the character table contains the characters themselves. Thus, eachrow constitutes the characters of an irreducible representation, and thenumber of rows gives us the number of irreducible representations ofthe particular point group. The irreducible representations have someimportant and useful properties:1. The sum of the squares of the dimensions of all irreducible representations in a group is equal to the order of the group. The dimension of an irreducible representation is simply the dimension ofany of its matrices, which is the number of rows or columns of thematrix.
Since the identity operation always leaves the moleculesunchanged, its representation is a unit matrix. The character ofa unit matrix is equal to the number of rows or columns of thatmatrix, as is demonstrated on the next page:Table 4-3. A Preliminary Character Table for the C2h Point GroupC2hEC2ih⌫1⌫2⌫3⌫411111−11−111−1−11−1−111924 Helpful Mathematical Tools⎡1⎣E= 001E=0 E= 12.3.4.5.01001⎤00⎦1character = 1 + 1 + 1 = 3character = 1 + 1 = 2character = 1From this it follows that the character under E is alwaysthe dimension of the given irreducible representation.
Theone-dimensional representations are nondegenerate and the two- orhigher-dimensional representations are degenerate. The meaningof degeneracy will be discussed in Chapter 6.The sum of the squares of the absolute values of characters of anyirreducible representation in a group is equal to the order of thegroup.The sum of the products of the corresponding characters (or onecharacter with the conjugate of another in case of imaginary characters) of any two different irreducible representations of the samegroup is zero.The characters of all matrices belonging to operations in the sameclass are identical in a given irreducible representation.The number of irreducible representations of a group is equal tothe number of classes of that group.Let us check these rules on the C2h character table given above.
Allfour irreducible representations have 1 as their character under E, soall of them are one-dimensional. Applying rule 1,12 + 12 + 12 + 12 = 4This is, indeed, the order of the group since there are four symmetryoperations in C2h . Let us check rule 2 with the ⌫2 representation:12 + (−1)2 + 12 + (−1)2 = 44.5.
Character Tables and Properties of Irreducible Representations193This is, again, the order of the group. Let us form the sum of theproducts of ⌫3 and ⌫4 according to rule 3:1 · 1 + 1 · (−1) + (−1) · (−1) + (−1) · 1 = 0Since all four symmetry elements in C2h stand by themselves, rule4 cannot be checked with this point group. Finally, the number of irreducible representations is four just as the number of classes, accordingto rule 5.Table 4-4 shows a preliminary character table for the C3v pointgroup. The complete set of symmetry operations is listed in the upperrow. Clearly, some of them must belong to the same class sincethe number of irreducible representations is 3 and the number ofsymmetry operations is 6.
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