M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 28
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These tables contain the products of the elements of a group. An example is shown in Table 4-1, forthe C2v point group. Here each element of the group, that is, each4.1. Groups173Table 4-1. Group Multiplication Table for the C2v Point GroupC2EC2vEC2vEC2vvC2EvEC2C2Evsymmetry operation, is listed only once in the initial row at the topand in the initial column at the far left.
In forming the product of anytwo elements, one belonging to the row and the other to the column,the order of the application of the elements is strictly defined. First, theelement in the top row is applied, followed by the application of theelement in the far left column. The result is found at the intersectionof the corresponding column and row. Any one of the results is alsoa symmetry operation belonging to the C2v point group.
In fact, eachrow and each column in the field of the results is a rearranged list ofthe initial operations, but no two rows or two columns may be identical. From the C2v multiplication table, it is seen that the inverse operation of C2 is C2 , since their intersection is E; similarly, the inverseoperation of v is v in this group.The multiplication table of the C3v point group is compiled inTable 4-2. Here,C3 · C3 = C32means two successive applications of the threefold rotation. Applyingit once yields a 120◦ rotation, while C32 corresponds to a 240◦ rotationTable 4-2. Group Multiplication Table for the C3v Point GroupC3EC3C32vvEC3C32vvEC3C32vvC3C32EvvC32EC3vvvvEC32C3vvC3EC32vvC32C3E1744 Helpful Mathematical Toolsaltogether.
Accordingly, for example, the meaning of C52 is a rotationby 2 · (360◦ /5) = 144◦ .The number of elements in a group is called the order of the group.Its conventional symbol is h. The group multiplication tables showthat h = 4 for the C2v point group and h = 6 for C3v .A group may be divided into two kinds of subunit: subgroupsand classes. A subgroup is a smaller group within a group that stillpossesses the four fundamental properties of a group. The identityoperation, E, is always a subgroup by itself, and it is also a member ofall other possible subgroups.A class is a complete set of elements, in our case symmetry operations, of the group that are conjugate to one another (in mathematicsthey are usually called conjugacy class). Elements A and B of a groupare conjugates, if there is some group element, Z, for whichB = Z −1 · A · ZDesignating a conjugate B to a symmetry operation A is also calleda similarity transformation.
B is a similarity transform of A by Z,or, in other words, A and B are conjugates. Elements belong to oneclass if they are conjugate to one another. The inverse operation canbe applied with the aid of the multiplication table and rule 4 givenabove,Z −1 · Z = Z · Z −1 = ETo find out what operations belong to the same class within agroup, all possible similarity transformations in the group have to beperformed. Let us work this out for the C3v point group and begin withthe identity operation.
Since E commutes with any other elements Z(see under rule 2 above), we haveZ −1 · E · Z = Z −1 · Z · E = E · E = Efor all elements in the class. Consequently, E is not conjugate with anyother element, and it always forms a class by itself. This is true for allother point groups as well.4.1. Groups175Consider now v :E −1 · (v · E) = E −1 · v = vC3−1 · (v · C3 ) = C3−1 · v = C32 · v = v(C32 )−1 · (v · C32 ) = (C32 )−1 · v = C3 · v = v−1−1v · (v · v ) = v · E = v · E = v−1· (v · v ) = −1· C3 = v · C3 = vvv−1· (v · v ) = −1· C32 = v · C32 = vvvWe have performed all possible similarity transformations for theoperation v .
As a result, it is seen that the three operations expressingvertical mirror symmetry belong to the same class. We could reach thesame conclusion by similarity transformations on either of the othertwo v operations.Next let us examine C3 :E −1 · (C3 · E) = E −1 · C3 = E · C3 = C3C3−1 · (C3 · C3 ) = C3−1 · C32 = C32 · C32 = C3(C32 )−1 · (C3 · C32 ) = (C32 )−1 · E = C3 · E = C3−12−1v · (C 3 · v ) = v · v = v · v = C 3−1· (C3 · v ) = −1· v = v · v = C32vv−1· (C3 · v ) = −1· v = v · v = C32vvAccording to these transformations, C3 and C32 are conjugates andthus belong to the same class.The order of a class is defined as the number of elements in theclass.
For example, the order of the class of the reflection operationsin C3v is 3, and the order of the class of the rotation operations is 2.The order of a class, or a subgroup, is an integral divisor of the orderof the group.The mathematical handling of the symmetry operations is done bymeans of matrices.1764 Helpful Mathematical Tools4.2. MatricesA matrix is a rectangular array of numbers, or symbols for numbers.These elements are put between square brackets.
A numericalexample of a matrix is shown here:⎡⎤3102⎣570 −3⎦00 −21Generally a matrix has m rows and n columns:⎡a11⎢ a21⎢⎢ ·⎢⎢ ·⎣ ·am1a12a22······am2···⎤a1na2n ⎥⎥· ⎥⎥· ⎥· ⎦amnThe above matrix may be represented by a capital letter A. Anothernotation is [ai j ]. The symbol ai j represents the matrix elementstanding in the ith row and the jth column. The number of rows ism, and the number of columns is n, and 1 ≤ i ≤ m and 1 ≤ j ≤ n.There are some special matrices important for our discussion. Asquare matrix has equal numbers of rows and columns. According tothe general notation, a matrix [ai j ] is a square matrix if m = n. Thedimension of a square matrix is the number of its rows or columns.A special square matrix is the unit matrix, in which all elementsalong the top-left-to-bottom-right diagonal are 1 and all the otherelements are zero.
The short notation for a unit matrix is E. Someunit matrices are presented here:⎡⎤1 0 0 0 0⎡⎤⎢0 1 0 0 0⎥1 0 0⎢⎥1 0⎢0 0 1 0 0⎥⎣0 1 0⎦⎢⎥0 1⎣0 0 0 1 0⎦0 0 10 0 0 0 14.2. Matrices177A column matrix consists of only one column. Column matrices areused to represent vectors.
A vector is characterized by its length anddirection. A vector in three-dimensional space is shown in Figure 4-4.If one end of the vector is at the origin of the Cartesian coordinatesystem, then the three coordinates of its other end fully describe thevector. These three Cartesian coordinates can be written as a columnmatrix:⎡ ⎤x1⎣ y1 ⎦z1Thus, this column matrix represents the vector.While column matrices are used to represent vectors, squarematrices are used to represent symmetry operations. Performing asymmetry operation on a vector is actually a geometrical transformation.
How can these geometrical transformations be translated intomatrix “language”? Consider a specific example and see how thesymmetry operations of the Cs symmetry group can be applied to thevector of Figure 4-4. For a matrix representation, we first write (orusually just imagine) the coordinates of the original vector in the topFigure 4-4. Representation of a vector in three-dimensional space.1784 Helpful Mathematical Toolsrow and the coordinates of the vector resulting from the symmetryoperation in the left-hand column:x1 y1 z 1⎡⎤resultant vectorx1⎢y1 ⎢⎣z 1← original vector⎥⎥⎦Then we examine the effect of the symmetry operation in detail. Ifa coordinate is transformed into itself, 1 is placed into the intersectionposition, and if it is transformed into its negative self, –1 is put intothe intersection.
Both these positions will be along the diagonal ofthe matrix. If a coordinate is transformed into another coordinate orinto the negative of this other coordinate, 1 or –1 is placed into theintersection position, respectively. These intersection positions will beoff the matrix diagonal.There are two symmetry operations in the Cs point group, E and h .The identity operation, E, does not change the position of the vectorso it can be represented by a unit matrix.⎡⎤ ⎡ ⎤ ⎡ ⎤1 0 0x1x1⎣0 1 0⎦ · ⎣ y1 ⎦ = ⎣ y1 ⎦z1z10 0 1Accordingly,E · v1 = v1If the matrix elements are aij and the vector components are bj , thenthe components of the product vector ci are given byai j · b j .ci =jTo get the first member of the resulting matrix, all the elements ofthe first row of the square matrix are multiplied by the consecutivemembers of the column matrix and then added together. To get the4.2.
Matrices179second member, the same procedure is followed with the second rowof the square matrix, and so on, as shown below:⎤ ⎡ ⎤⎡⎤ ⎡ ⎤ ⎡1 · x1 + 0 · y1 + 0 · z 1x11 0 0x1⎣0 1 0⎦ · ⎣ y1 ⎦ = ⎣0 · x1 + 1 · y1 + 0 · z 1 ⎦ = ⎣ y1 ⎦z10 · x1 + 0 · y1 + 1 · z 1z10 0 1The other symmetry operation of the Cs point group is the horizontal reflection (see Figure 4-5).
In matrix language this operationcan be written as follows:x1y1z 1⎡x11⎢⎣00y1010Ez1⎤ ⎡ ⎤ ⎡⎤ ⎡⎤x101 · x1 + 0 · y1 + 0 · z 1x1⎥ ⎢ ⎥ ⎢⎥ ⎢⎥0 ⎦ · ⎣ y1 ⎦ = ⎣ 0 · x1 + 1 · y1 + 0 · z 1 ⎦ = ⎣ y1 ⎦z10 · x1 + 0 · y1 + (−1) · z 1−z 1−1·v1=v2It often happens that the coordinates are not transformed simplyinto each other by a symmetry operation. Trigonometric relationsmust be used to express, for instance, the consequences of three-foldrotation.Figure 4-5.
Reflection of a vector by a horizontal mirror plane.1804 Helpful Mathematical ToolsFigure 4-6. Rotation of a vector by an angle ␣ in the xy plane.Figure 4-6 illustrates a vector rotated by an angle ␣ in the xy plane.The coordinates of the rotated vector are related to the coordinatesof the original vector in the following way (r is the length of thevector,  is an auxiliary angle shown in Figure 4-6, and the rotation isanticlockwise):x1 = r · cos x2 = r · cos(␣ + )andandy1 = r · sin (4-1)y2 = r · sin(␣ + )(4-2)Utilizing the trigonometric expressions:cos(␣ + ) = cos ␣ · cos  − sin ␣ · sin (4-3a)sin(␣ + ) = sin ␣ · cos  + cos ␣ · sin (4-3b)and substituting Eqs.
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