结构化学 分子对称性
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而垂直于Cn轴通过交点的平面内必有n个C2轴
对称元素的组合-2
2、两个镜面的组合
交角为2pi/2n的两个镜面相交，则其交线必为n
次轴Cn ； Cn轴和通过它的镜面组合，一定存在n个镜面，
相邻镜面的夹角为2pi/2n
对称元素的组合-3 3、偶次旋转轴和与它垂直的镜面的组合
一个偶次旋转轴和与它垂直的镜面的组合，必定在其 交点上出现对称中心---〉一个偶次旋转轴和对称中心
n-重映轴 n-fold improper axis
of rotation
n-重反轴 n-fold improper axis
of rotation
对称操作
Symmetry operation
不变 nothing
What does it do Nothing
按对称中心反演 projects through the center an
Cn，Cnh， Cnv， Cni， Sn， Dn， Dnh， Dnd, T, Th, Td, O, Oh, I, Ih
Point groups
Point groups are a way of classifying molecules in terms of their internal symmetry. Molecules can have many symmetry operations that result into indistinguishable configurations. Different collections of symmetry operations are organized into groups. These 11 groups were developed by Schoenflies.
axis
rotates counterclockwise 360o/n degrees about the axis
and then projects through5 the center an equal distance
a)具有对称中心的
b)没有对称中心的
1 0 0 i0 1 0
0 0 1
inE, n iseven i, n isodd
组合，必有垂直于轴的镜面；，，，
群的定义
1、封闭性： A ,B G ,A B C , C G
2、主操作：
A E E A A
3、逆操作： A G , A 1 G ,A 1 A A 1 A E
4、结合律：
A(B)C (A)B C
ห้องสมุดไป่ตู้
群的实例
群的乘法表 规则：先行后列，（列行）
分子点群的分类
by reflection
绕轴旋转后按对 称中心反演
rotation followed by inversion
rotates counterclockwise 360o/n degrees about the axis
and then reflects across a
plane perpendicular to the
Cn:
only a Cn center of symmetry.
Example of C2: hydrogen peroxide (not coplanar)
Cnv: only n-fold axis and n vertical (or dihedral) mirror planes.
h, :v,
d,
Cn,(horizon)tal containsCn(vertic)al containsCn and ...
σ垂直于主轴
σ通过主轴
σ通过主轴，平分两副轴（C2轴）的夹角
旋转反映操作和映轴
旋转反演操作和反轴
对称元素的组合-1
1、两个旋转轴的组合
交角为2pi/2n的两个C2轴，在其交点上必定出 现一个垂直于这两个C2轴的Cn轴；
inversion
equal distance
绕轴旋转 rotation
rotates counterclockwise 360o/n degrees about the axis
通过镜面反映 reflection
reflects across a plane
绕轴旋转后反映 rotation followed
第四章 分子的对称性
第七章 晶体结构的对称性
Symmetry is important in quantum mechanics for determining molecular structure and for interpreting spectroscopic information.
In addition of being used to simplify calculations, two properties directly depend on symmetry: optical activity and dipole moments.
We consider equilibrium configurations, with the atoms in their mean positions.
a)氨分子（NH3）的三重轴 b)水分子（H2O）的二重轴
x x '
C
k n
y
y
'
z z '
co2sk(/n) sin2k(/n) 0
Cnk sin2k(/n) co2sk(/n) 0
0
0
1
反映操作和镜面
1 0 0
xy0 1 0
0 0 1
nE, n iseven , n isodd
C1:
only identity. Example: CHBrClF
Cs:
only a reflection plane. Example: CH2BrCl
Ci:
only a center of symmetry. Example: staggered 1,2-dibromo-1,2-
dichloroethane.
符号标记 Label
Eˆ ˆi Cˆ n ˆ Sˆ n
Iˆ n
对称元素 Symmetry element
恒等操作 identity
对称中心 center of symmetry or inversion center
n-重旋转轴 n-fold proper axis of
rotation 镜面
plane of symmetry
对称元素的组合-2
2、两个镜面的组合
交角为2pi/2n的两个镜面相交，则其交线必为n
次轴Cn ； Cn轴和通过它的镜面组合，一定存在n个镜面，
相邻镜面的夹角为2pi/2n
对称元素的组合-3 3、偶次旋转轴和与它垂直的镜面的组合
一个偶次旋转轴和与它垂直的镜面的组合，必定在其 交点上出现对称中心---〉一个偶次旋转轴和对称中心
n-重映轴 n-fold improper axis
of rotation
n-重反轴 n-fold improper axis
of rotation
对称操作
Symmetry operation
不变 nothing
What does it do Nothing
按对称中心反演 projects through the center an
Cn，Cnh， Cnv， Cni， Sn， Dn， Dnh， Dnd, T, Th, Td, O, Oh, I, Ih
Point groups
Point groups are a way of classifying molecules in terms of their internal symmetry. Molecules can have many symmetry operations that result into indistinguishable configurations. Different collections of symmetry operations are organized into groups. These 11 groups were developed by Schoenflies.
axis
rotates counterclockwise 360o/n degrees about the axis
and then projects through5 the center an equal distance
a)具有对称中心的
b)没有对称中心的
1 0 0 i0 1 0
0 0 1
inE, n iseven i, n isodd
组合，必有垂直于轴的镜面；，，，
群的定义
1、封闭性： A ,B G ,A B C , C G
2、主操作：
A E E A A
3、逆操作： A G , A 1 G ,A 1 A A 1 A E
4、结合律：
A(B)C (A)B C
ห้องสมุดไป่ตู้
群的实例
群的乘法表 规则：先行后列，（列行）
分子点群的分类
by reflection
绕轴旋转后按对 称中心反演
rotation followed by inversion
rotates counterclockwise 360o/n degrees about the axis
and then reflects across a
plane perpendicular to the
Cn:
only a Cn center of symmetry.
Example of C2: hydrogen peroxide (not coplanar)
Cnv: only n-fold axis and n vertical (or dihedral) mirror planes.
h, :v,
d,
Cn,(horizon)tal containsCn(vertic)al containsCn and ...
σ垂直于主轴
σ通过主轴
σ通过主轴，平分两副轴（C2轴）的夹角
旋转反映操作和映轴
旋转反演操作和反轴
对称元素的组合-1
1、两个旋转轴的组合
交角为2pi/2n的两个C2轴，在其交点上必定出 现一个垂直于这两个C2轴的Cn轴；
inversion
equal distance
绕轴旋转 rotation
rotates counterclockwise 360o/n degrees about the axis
通过镜面反映 reflection
reflects across a plane
绕轴旋转后反映 rotation followed
第四章 分子的对称性
第七章 晶体结构的对称性
Symmetry is important in quantum mechanics for determining molecular structure and for interpreting spectroscopic information.
In addition of being used to simplify calculations, two properties directly depend on symmetry: optical activity and dipole moments.
We consider equilibrium configurations, with the atoms in their mean positions.
a)氨分子（NH3）的三重轴 b)水分子（H2O）的二重轴
x x '
C
k n
y
y
'
z z '
co2sk(/n) sin2k(/n) 0
Cnk sin2k(/n) co2sk(/n) 0
0
0
1
反映操作和镜面
1 0 0
xy0 1 0
0 0 1
nE, n iseven , n isodd
C1:
only identity. Example: CHBrClF
Cs:
only a reflection plane. Example: CH2BrCl
Ci:
only a center of symmetry. Example: staggered 1,2-dibromo-1,2-
dichloroethane.
符号标记 Label
Eˆ ˆi Cˆ n ˆ Sˆ n
Iˆ n
对称元素 Symmetry element
恒等操作 identity
对称中心 center of symmetry or inversion center
n-重旋转轴 n-fold proper axis of
rotation 镜面
plane of symmetry