厦门大学：《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 3 Molecular symmetry and symmetry point group

For example These have a center of inversion i. 0=C=0 These do not have a center of inversion. H Si H H H

These do not have a center of inversion. These have a center of inversion i . For example

>Inverts all atoms through the centre of the object Inversion 4 3 3 Center of inversion 6 >Its matrix representation -1 0 0 0 -1 0 0 0 -1

    0 0 1 0 1 0 1 0 0              zyx zyx zyx i 0 0 1 0 1 0 1 0 0 Its matrix representation Inverts all atoms through the centre of the object

3)Rotation and the n-fold rotation axis(C) Rotation about an n-fold axis(rotation through 3600/n)is denoted by the symbol C Example:Rotation of trigonal planer BF3. One three-fold(C3)rotation axes. (o=2π/3) C3 C l20°rotation F 120°rotation → → The principle rotation axis is the axis of the highest fold

• Example: Rotation of trigonal planer BF 3. – One three-fold (C 3) rotation axes. ( =2 /3) C 3 1 C 3 2 Rotation about an n-fold axis (rotation through 360 o/n) is denoted by the symbol C n . 3) Rotation and the n-fold rotation axis (C n ) The principle rotation axis is the axis of the highest fold

The matrix representations: o(xy) x'=-rsin(30°+a)=-rsin30°cosa-rcos30°sinc =(1/2)x+(-V3/2)y 30°+C y'=rcos(30°+)=rcos30°cosa-rsin30°sinc 120° =(N3/2)x+(1/2)y O(xy) 2π 2π 5 0 0 3 X cos 3 sin x INLc C y sin c05 2π 21 y 0 y x-Y 20 20 2 0 0 1 1 4π 1 3 V3 cos sin 4π 0 0 y x2 1 2 y2 y sin 3 cos 3 0 y 0 y 2 2 z 0 25-20 1 20 1 z

The matrix representations: 30  ＋   （x,y ） （x’,y’ ） C 3 1 x y x’=-rsin(30  ＋ )= -rsin30 cos  - rcos30 sin  = (-1/2)x + (- 3/2)y y’=rcos(30  ＋ )= rcos30 cos  - rsin30 sin  = ( 3/2)x + (-1/2)y                                                                                                         z y x y x z y x z y x sin cos cos sin z y x C z y x 2 2 3 2 3 2 0 0 1 0 2 1 2 3 0 2 3 2 1 0 0 1 0 3 4 3 4 0 3 4 3 4 2 3 2 2 2                                                                                                            z y x y x z y x z y x sin cos cos sin z y x C z y x 2 2 3 2 3 2 0 0 1 0 2 1 2 3 0 2 3 2 1 0 0 1 0 3 2 3 2 0 3 2 3 2 1 3 1 1 1     120 

The matrix representations: Conditions: >The centre of mass of the molecule is located at the origin of the Cartesian Coordinate System >Principle axis is aligned with the z-axis C -x,-y】 cOSπ -S1nπ 0 0 0 y C, y sinπ COSπ 0 0 0 0 0 0 0 X,Y) cosa -sina 0 、2kπ sina cosa 0 a=- n 0 0

    0 0 1 sin cos 0 cos sin 0     k Cn nk  2  2 cos sin 0 1 0 0 sin cos 0 0 1 0 0 0 1 0 01 x x x xx y Cy y y y z z z zz                                                                           The matrix representations: C2 Cn （x,y） （-x,-y） x y Conditions:  The centre of mass of the molecule is located at the origin of the Cartesian Coordinate System  Principle axis is aligned with the z-axis