M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 29

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(4-3) and (4-1) into Eq. (4-2), we get:x2 = r · cos ␣ · cos ␤ − r · sin ␣ · sin ␤ = x1 · cos ␣ − y1 · sin ␣ (4-4a)y2 = r · sin ␣ · cos ␤ + r · cos ␣ · sin ␤ = x1 · sin ␣ + y1 · cos ␣ (4-4b)4.2. Matrices181The same equations in matrix formulation: − sin ␣xx· 1 = 2y1y2cos ␣cos ␣sin ␣The square matrix above is the matrix representation of a rotationthrough an angle ␣.Since matrices can be used to represent symmetry operations, theset of matrices representing all symmetry operations of a point groupwill be a representation of that group. Moreover, if a set of matricesforms a representation of a symmetry group, it will obey all therules of a mathematical group.

It will also obey the group multiplication table. Let the SO2 Cl2 molecule serve as an example again. Thismolecule belongs to the C2v point group and some of its symmetryoperations have already been illustrated in Figure 4-2.There are four operations in the C2v point group. The identity operation, E, leaves the molecule unchanged, so we can imagine that thecorresponding matrix will be a 5 × 5 unit matrix.The twofold rotation (C2 ) changes the positions of the two chlorineatoms and also the positions of the two oxygen atoms.

The sulfur atomremains in place. To construct the corresponding matrices the sameprocedure can be applied as used before with a vector. The originalnuclear positions of the molecule can be written (or just imagined)at the top row and the nuclear positions resulting from the symmetryoperation at the far left column. Thus the C2 operation will lead to thefollowing result:⎡1⎢0⎢⎢0⎢⎣00001000100000001⎤ ⎡ ⎤ ⎡ ⎤0S1S1⎢Cl2 ⎥ ⎢Cl3 ⎥0⎥⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥0⎥⎥ · ⎢Cl3 ⎥ = ⎢Cl2 ⎥1⎦ ⎣ O4 ⎦ ⎣ O5 ⎦O5O40␴v changes the positions of the two chlorines and leaves the otherthree atoms in place:1824 Helpful Mathematical Tools⎡1⎢0⎢⎢0⎢⎣00001000100000010⎤ ⎡ ⎤ ⎡ ⎤0S1S1⎢Cl2 ⎥ ⎢Cl3 ⎥0⎥⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥0⎥⎥ · ⎢Cl3 ⎥ = ⎢Cl2 ⎥0⎦ ⎣ O4 ⎦ ⎣ O4 ⎦O5O51Finally, ␴v changes the positions of the two oxygen atoms, andleaves the sulfur and the two chlorines in their original positions:⎡⎤ ⎡ ⎤ ⎡ ⎤1 0 0 0 0S1S1⎢0 1 0 0 0⎥ ⎢Cl2 ⎥ ⎢Cl2 ⎥⎢⎥ ⎢ ⎥ ⎢ ⎥⎢0 0 1 0 0⎥ · ⎢Cl3 ⎥ = ⎢Cl3 ⎥⎢⎥ ⎢ ⎥ ⎢ ⎥⎣0 0 0 0 1⎦ ⎣ O4 ⎦ ⎣ O5 ⎦O5O40 0 0 1 0Since each of these four 5 × 5 matrices represent one of thesymmetry operations of the C2v point group, the set of these four 5 × 5matrices will be a representation of this group.

They will also obey theC2v multiplication table. As was shown in Figure 4-2,␴v · C2 = ␴vThe corresponding matrix representations are the following:⎤⎤ ⎡1 0 0 0 00 0 01 0 0⎥ ⎢0 0 1 0 0⎥⎥⎥ ⎢0 0 0⎥ · ⎢0 1 0 0 0⎥ =0 1 0⎦ ⎣0 0 0 0 1⎦0 0 0 1 00 0 1␴vC2⎡⎤1·1 + 0·0 + 0·0 + 0·0 + 0·0 1·0 + 0·0 + 0·1 + 0·0 + 0·0 .

. .⎢0·1 + 0·0 + 1·0 + 0·0 + 0·0 0·0 + 0·0 + 1·1 + 0·0 + 0·0 . . .⎥⎢⎥⎢0·1 + 1·0 + 0·0 + 0·0 + 0·0 0·0 + 1·0 + 0·1 + 0·0 + 0·0 . . .⎥ =⎣0·1 + 0·0 + 0·0 + 1·0 + 0·0 0·0 + 0·0 + 0·1 + 1·0 + 0·0 . . .⎦0·1 + 0·0 + 0·0 + 0·0 + 1·0 0·0 + 0·0 + 0·1 + 0·0 + 1·0 . . .⎤⎡1 0 0 0 0⎢0 1 0 0 0 ⎥⎥⎢⎢0 0 1 0 0 ⎥⎣0 0 0 0 1 ⎦0 0 0 1 0␴v⎡1⎢0⎢⎢0⎣00001004.3. Representation of Groups183The multiplication is shown here in detail only for the first twocolumns of the resulting matrix. The elements of the product matrixare given by:cik =ai j · b jkjTo get the first member of the first row, all elements of the first rowof the first matrix are multiplied by the corresponding elements of thefirst column of the second matrix and the results are added. To getthe second member of the first row, all elements of the first row ofthe first matrix are multiplied by the corresponding members of thesecond column of the second matrix and the results are added, and soon.

To get the second-row members, the same procedure is repeatedwith the second-row members of the first matrix, and so on. It is alsopossible to visualize the second matrix as a series of column matricesand then consider the multiplication of each of these column matrices,one by one, by the first matrix.4.3. Representation of GroupsAny collection of quantities (or symbols) which obey the multiplication table of a group is a representation of that group. These quantities are the matrices in our examples showing how certain characteristics of a molecule behave under the symmetry operations of thegroup. The symmetry operations may be applied to various characteristics or descriptions of the molecule.

The particular descriptionto which the symmetry operations are applied forms the basis for arepresentation of the group. Generally speaking, any set of algebraicfunctions or vectors may be the basis for a representation of a group.Our choice of a suitable basis depends on the particular problem weare studying. After choosing the basis set, the task is to construct thematrices which transform the basis or its components according toeach symmetry operation. The most common basis sets in chemicalapplications are summarized in Section 4.11. Some of them will beused in the following discussion.1844 Helpful Mathematical ToolsLet us now work out the representation of a point group for avery simple basis.

We will choose just the changes, ⌬r1 and ⌬r2 ,of the two N–H bond lengths of the diimide molecule, N2 H2 (4-1).These two vectors may be used in the description of thestretching vibrations of the molecule. The molecular symmetry isC2h . Figure 4-7 helps to visualize the effects of the symmetry operations of this group on the selected basis. There are four symmetryoperations in the C2h point group, E, C2 , i, and ␴h .

E leaves thebasis unchanged, so the corresponding matrix representation is a unitmatrix: ⌬r11E·=⌬r20 0⌬r1⌬r1·=⌬r2⌬r21Figure 4-7. The four symmetry operations of the C2h point group applied to thetwo N–H bond length changes of the HNNH molecule.4.3. Representation of Groups185Both C2 and i interchange the two vectors, ⌬r1 “goes into” ⌬r2 andvice versa; ⌬r10 1⌬r1⌬r2C2 ·=·=1 0⌬r2⌬r2⌬r1 ⌬r10i·=⌬r21 1⌬r1⌬r2·=⌬r2⌬r10finally, ␴h leaves the molecule unchanged: 1⌬r1␴h ·=⌬r20 0⌬r1⌬r1·=⌬r2⌬r21With this basis the representation consists of four 2 × 2 matrices.Let us take now a more complicated basis, and consider all thenuclear coordinates of HNNH shown in Figure 4-8a. These are theso-called Cartesian displacement vectors and will be discussed inChapter 5 on molecular vibrations. Let us find the matrix representation of the ␴h operation (see Figure 4-8b).

The horizontal mirror planeleaves all x and y coordinates unchanged while all z coordinates will“go” into their negative selves. In matrix notation this is expressed inthe following way:⎡ ⎤ ⎡1x1⎢ y1 ⎥ ⎢ 0⎢ ⎥ ⎢⎢z1 ⎥ ⎢ 0⎢ ⎥ ⎢⎢x2 ⎥ ⎢ 0⎢ ⎥ ⎢⎢ y2 ⎥ ⎢ 0⎢ ⎥ ⎢⎢z2 ⎥ ⎢ 0⎥ ⎢␴h · ⎢⎢x3 ⎥ = ⎢ 0⎢ ⎥ ⎢⎢ y3 ⎥ ⎢ 0⎢ ⎥ ⎢⎢z3 ⎥ ⎢ 0⎢ ⎥ ⎢⎢x4 ⎥ ⎢ 0⎢ ⎥ ⎢⎣ y4 ⎦ ⎣ 00z40 01 00 −10 00 00 00 00 00 00 00 00 00001000000000 00 00 00 01 00 −10 00 00 00 00 00 00000001000000 00 00 00 00 00 00 01 00 −10 00 00 0000000000100⎤ ⎡ ⎤ ⎡⎤x10 0x1⎢⎥⎢⎥0 0 ⎥ ⎢ y1 ⎥⎥ ⎢ y1 ⎥⎢ z 1 ⎥ ⎢ −z 1 ⎥0 0⎥⎥ ⎢ ⎥ ⎢⎥⎢ ⎥ ⎢⎥0 0⎥⎥ ⎢ x2 ⎥ ⎢ x2 ⎥⎢ y2 ⎥ ⎢ y2 ⎥0 0⎥⎥ ⎢ ⎥ ⎢⎥⎢ ⎥ ⎢⎥0 0⎥⎥ · ⎢ z 2 ⎥ = ⎢ −z 2 ⎥⎢⎥⎢⎥0 0 ⎥ ⎢ x3 ⎥ ⎢ x3 ⎥⎥⎢ ⎥ ⎢⎥0 0⎥⎥ ⎢ y3 ⎥ ⎢ y3 ⎥⎢ z 3 ⎥ ⎢ −z 3 ⎥0 0⎥⎥ ⎢ ⎥ ⎢⎥⎢ ⎥ ⎢⎥0 0⎥⎥ ⎢ x4 ⎥ ⎢ x4 ⎥1 0 ⎦ ⎣ y4 ⎦ ⎣ y4 ⎦0 −1z4−z 41864 Helpful Mathematical ToolsFigure 4-8. (a) Cartesian coordinates as basis for a representation; (b) Effect of ␴h ;(c) Effect of C2 .Take one more operation, the C2 rotation (Figure 4-8c).

This operation introduces the following changes:x1 , y1 , and z 1 to −x4 , −y4 , and z 4 ,x2 , y2 , and z 2 to −x3 , −y3 , and z 3 ,x3 , y3 , and z 3 to −x2 , −y2 , and z 2 ,x4 , y4 , and z 4 to −x1 , −y1 , and z 1 .4.3. Representation of Groups187In matrix notation:⎡ ⎤ ⎡0 0x1⎢ y1 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢z1 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢x2 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢ y2 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢z2 ⎥ ⎢ 0 0⎥ ⎢C2 · ⎢⎢x3 ⎥ = ⎢ 0 0⎢ ⎥ ⎢⎢ y3 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢z3 ⎥ ⎢ 0 0⎢ ⎥ ⎢⎢x ⎥ ⎢ −1 0⎢ 4⎥ ⎢⎣ y ⎦ ⎣ 0 −140 0z40 0 00 0 00 0 00 0 00 0 00 0 00 −1 00 0 −10 0 00 0 00 0 01 0 00 0 00 0 00 0 00 −1 00 0 −10 0 00 1 00 0 11 0 00 0 00 0 00 0 00 −1 00 0 −10 0 00 0 00 0 01 0 00 0 00 0 00 0 00 0 00 0 00 0 0001000000000⎤ ⎡ ⎤ ⎡⎤x1−x4⎥ ⎢ y1 ⎥ ⎢ −y4 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢z1 ⎥ ⎢ z4 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢x2 ⎥ ⎢ −x3 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢ y2 ⎥ ⎢ −y3 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢z2 ⎥ ⎢ z3 ⎥⎥·⎢ ⎥=⎢⎥⎥ ⎢x3 ⎥ ⎢ −x2 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢ y3 ⎥ ⎢ −y2 ⎥⎥ ⎢ ⎥ ⎢⎥⎥ ⎢z ⎥ ⎢ z ⎥⎥ ⎢ 3⎥ ⎢ 2 ⎥⎥ ⎢x ⎥ ⎢ −x ⎥⎥ ⎢ 4⎥ ⎢ 1 ⎥⎦ ⎣ y ⎦ ⎣ −y ⎦41z4z1Considering all four symmetry operations of the C2h point group,the complete representation of the displacement coordinates of HNNHas basis consists of four 12 × 12 matrices.

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