《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 3 Molecular Symmetry and Point Group Part A

The principal rotation axis is the axis of the highest fold一a6H

C5 The principal rotation axis is the axis of the highest fold. C6

The matrix representations of rotations around a C, axis:Conditions:> Principal axis is aligned with the z-axisA C, axis gives rise to n unique rotationaloperations labeled as C,m (m = 1, 2,..., n)AngleofallowedMatrix of allowed rotationsrotations0-sinαcos a2m元α =m0Cnsina二cos aαnn010(m = 1,2,...,n)

The matrix representations of rotations around a Cn axis: Cn Conditions:  Principal axis is aligned with the z-axis A Cn axis gives rise to n unique rotational operations labeled as Cn m (m = 1, 2,., n).             0 0 1 0 0     sin cos cos sin m Cn n m  2  (m = 1,2,.,n) Matrix of allowed rotations Angle of allowed rotations:

The matrix representation of rotational operations: e.g., C,mCn IIzAllowed rotations: α = 2mn (m =1,2,...,n)x,=-rcosB=rcos(α+A)=rcosAcosα-rsinAsinaB=元-(A+α=xcosα-ysinα(xy)yi=rsinB=xsinα+ycosαBZi=zα0-sinαcosax大xym0sinacosayV00Z?(xy)0-sinαcosαx=rcosA0sinαcosa00y=rsin

The matrix representation of rotational operations: e.g., Cn m A (x,y) (x1 ,y1 ) Cn ||z X Y              0 0 1 0 0     sin cos cos sin m Cn                                            z y x z y x C z y x m n 0 0 1 0 0 1 1 1     sin cos cos sin x y r r  x = rcosA y = rsinA B      cos sin cos cos sin sin cos cos( ) x y r A r A x r B r A     1     B  ( A ) y1  rsinB  x sin  y cos Allowed rotations:  = 2mn (m =1,2,.,n) z1= z

Reflection and the Mirror plane (o)>If reflection of an object throughaplane produces anindistinguishable configuration, that plane is a plane of symmetry(mirrorplane,)(x, -y, z)(x,y,z)?100xTx00O-10一OO1XZxz000

 If reflection of an object through a plane produces an indistinguishable configuration, that plane is a plane of symmetry (mirror plane, s). 4) Reflection and the Mirror plane (s)             0 0 1 0 1 0 1 0 0 s xz                                             z y x z y x z y x xz 0 0 1 0 1 0 1 0 0 s （x, -y, z） （x,y,z） z y x z sxz x y sxz syz

Likewise, we have000x0xX10000ayVxyxy000Z000xXOOVZ0

            0 0 1 0 1 0 1 0 0 s xy                                             z y x z y x z y x xy 0 0 1 0 1 0 1 0 0 s Likewise, we have            0 0 1 0 1 0 1 0 0 s yz                                           z y x z y x z y x yz 0 0 1 0 1 0 1 0 0 s