下の表のいちばん右側の列は、その対称性を持つ分子の振動(x,y,z)および回転(Rx,Ry,Rz)の既約表現を示す。括弧の中の二つ以上の表示は縮退していることを示す。括弧なしで二つ以上の表示をしているものは、必ずしも縮退が存在しないことを示している。
|
C1=1 |
E |
|
|
A1 |
1 |
all |
|
Ci=S2=-1 |
E |
i |
|
|
Ag |
1 |
1 |
Rx, Ry, Rz, x2, y2, z2, xy, xz, yz |
|
Au |
1 |
-1 |
x, y, z |
|
Cs=C1h |
E |
sigmah |
|
|
A’ |
1 |
1 |
x, y, Rz, x2, y2, z2, xy |
|
A’’ |
1 |
-1 |
z, Rz, Ry, yz, xz |
|
C2=2 |
E |
C2 |
|
|
A |
1 |
1 |
z, Rz, x2, y2, z2, xy |
|
B |
1 |
-1 |
x, y, Rx, Ry, yz, xz |
|
C2v=mm2 |
E |
C2 |
sigmav |
sigmav’ |
|
|
A1 |
1 |
1 |
1 |
1 |
z, x2, y2, z2 |
|
A2 |
1 |
1 |
-1 |
-1 |
Rz, xy |
|
B1 |
1 |
-1 |
1 |
-1 |
x, Ry, xz |
|
B2 |
1 |
-1 |
-1 |
1 |
y, Rx, yz |
|
C2h=2/m |
E |
C2 |
i |
sigmah |
|
|
Ag |
1 |
1 |
1 |
1 |
Rz, x2, y2, z2, xy |
|
Bg |
1 |
-1 |
1 |
-1 |
Rx, Ry, xz, yz |
|
Au |
1 |
1 |
-1 |
-1 |
z |
|
Bu |
1 |
-1 |
-1 |
1 |
x, y |
|
D2=222 |
E |
C4 |
C2 |
C43 |
|
|
A |
1 |
1 |
1 |
1 |
x2, y2, z2 |
|
B1 |
1 |
1 |
-1 |
-1 |
z, Rz, xy |
|
B1 |
1 |
-1 |
1 |
-1 |
y, Ry, xz |
|
B2 |
1 |
-1 |
-1 |
1 |
x, Rx, yz |
|
D2h=2/mmm |
E |
C2 |
C2’ |
C2’’ |
i |
sigma(xy) |
sigma’(yz) |
sigma’’(xz) |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2, y2, z2 |
|
B1g |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
Rz, xy |
|
B2g |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
Ry, xz |
|
B3g |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
Rx, yz |
|
Au |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
|
|
B1u |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
z |
|
B2u |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
y |
|
B3u |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
x |
|
D2d=-42m |
E |
2S4 |
C2 |
2C2’ |
2sigmad |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1 |
1 |
-1 |
1 |
1 |
-1 |
x2-y2 |
|
B2 |
1 |
-1 |
1 |
-1 |
1 |
z, xy |
|
E |
2 |
0 |
-2 |
0 |
0 |
(x,y), (Rx,Ry), (xz,yz) |
|
C3=3 |
E |
C3 |
C32 |
|
|
A |
1 |
1 |
1 |
z, Rz, x2+y2, z2 |
|
E |
{1,1} |
{e,e*} |
{e*,e} |
(x,y), (Rx,Ry), (x2-y2,xy), (xz,yz) |
|
C3v=3m |
E |
2C3 |
3sigmav |
|
|
A1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2 |
1 |
1 |
-1 |
Rz |
|
E |
2 |
-1 |
0 |
(x,y), (Rx, Ry), (x2-y2,xy),(yz,zx) |
|
C3h=-6 |
E |
C3 |
C32 |
sigmah |
S3 |
S35 |
|
|
A’ |
1 |
1 |
1 |
1 |
1 |
1 |
Rz, x2+y2, z2 |
|
E’ |
{1,1} |
{e,e*} |
{e*,e} |
{1,1} |
{e,e*} |
{e*,e} |
(x,y), (x2-y2,xy) |
|
A’’ |
1 |
1 |
1 |
-1 |
-1 |
-1 |
Z |
|
E’’ |
{1,1} |
{e,e*} |
{e*,e} |
{-1,-1} |
{-e,-e*} |
{-e*,-e} |
(Rx,Ry), (xz, yz) |
|
S6=-3 |
E |
C3 |
C32 |
i |
S65 |
S6 |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
Rz, x2+y2, z2 |
|
Eg |
{1,1} |
{e,e*} |
{e*,e} |
{1,1} |
{e,e*} |
{e*,e} |
(Rx,Ry), (x2-y2,xy), (xz,yz) |
|
Au |
1 |
1 |
1 |
-1 |
-1 |
-1 |
Z |
|
Eu |
{1,1} |
{e,e*} |
{e*,e} |
{-1,-1} |
{-e,-e*} |
{-e*,-e} |
(x,y) |
|
D3=32 |
E |
2C3 |
3C2 |
|
|
A1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2 |
1 |
1 |
-1 |
Rz |
|
E |
2 |
-1 |
0 |
(x,y), (Rx,Ry),(x2-y2,xy),(yz,zx) |
|
D3d=-3m |
E |
2C3 |
3C2 |
i |
2S6 |
3sigmad |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2g |
1 |
1 |
-1 |
1 |
1 |
-1 |
Rz |
|
Eg |
2 |
-1 |
0 |
2 |
-1 |
0 |
(Rx,Ry), (x2-y2,xy), (xz,yz) |
|
A1u |
1 |
1 |
1 |
-1 |
-1 |
-1 |
|
|
A2u |
1 |
1 |
-1 |
-1 |
-1 |
1 |
z |
|
Eu |
2 |
-1 |
0 |
-2 |
1 |
0 |
(x,y) |
|
D3h=-6m2 |
E |
2C3 |
3C2 |
sigmah |
2S3 |
3sigmav |
|
|
A1’ |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2’ |
1 |
1 |
-1 |
1 |
1 |
-1 |
Rz |
|
E’ |
2 |
-1 |
0 |
2 |
-1 |
0 |
(x,y),(x2-y2,xy) |
|
A1’ ’ |
1 |
1 |
1 |
-1 |
-1 |
-1 |
|
|
A2’ ’ |
1 |
1 |
-1 |
-1 |
-1 |
1 |
z |
|
E’ ’ |
2 |
-1 |
0 |
-2 |
1 |
0 |
(Rx,Ry), (yz,zx) |
|
C4=4 |
E |
C4 |
C2 |
C43 |
|
|
A |
1 |
1 |
1 |
1 |
z, Rz, x2+y2, z2 |
|
B |
1 |
-1 |
1 |
-1 |
x2-y2, xy |
|
E |
{1,1} |
{i,-i} |
{-1,1} |
{-i,i} |
(x,y), (Rx,Ry), (xz,yz) |
|
C4h=4/m |
E |
C4 |
C42 |
C43 |
i |
S43 |
sigmah |
S4 |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
Rz,x2+y2,z2 |
|
Bg |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
x2-y2,xy |
|
Eg |
{1,1} |
{i,-i} |
{-1,-1} |
{-i,i} |
{1,1} |
{i,-i} |
{-1,-1} |
{-I,i} |
(Rx,Ry),(yz,zx) |
|
Au |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
Z |
|
Bu |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
|
|
Eu |
{1,1} |
{i,-i} |
{-1,-1} |
{-i,i} |
{-1,-1} |
{-i,i} |
{1,1} |
{i,-i} |
(x,y) |
|
C4v=4mm |
E |
2C4 |
C2 |
2sigmav |
2sigmad |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
z, x2+y2, z2 |
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1 |
1 |
-1 |
1 |
1 |
-1 |
x2-y2 |
|
B2 |
1 |
-1 |
1 |
-1 |
1 |
xy |
|
E |
2 |
0 |
-2 |
0 |
0 |
(x,y), (Rx,Ry), (xz, yz) |
|
D4=422 |
E |
2C4 |
C2 |
2C2’ |
2C2’’ |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
z, Rz |
|
B1 |
1 |
-1 |
1 |
1 |
-1 |
x2-y2 |
|
B2 |
1 |
-1 |
1 |
-1 |
1 |
xy |
|
E |
2 |
0 |
-2 |
0 |
0 |
(x,y), (Rx,Ry), (xz,yz) |
|
D4h=4/mmm |
E |
2C4 |
C2 |
2C2’ |
2C2’’ |
i |
2S4 |
sigmah |
2sigmav |
2sigmad |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2g |
1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1g |
1 |
-1 |
1 |
1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
(x,y), (x2-y2,xy) |
|
B2g |
1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
1 |
-1 |
1 |
xy |
|
Eg |
2 |
0 |
-2 |
0 |
0 |
2 |
0 |
-2 |
0 |
0 |
(Rx,Ry), (xz,yz) |
|
A1u |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
|
|
A2u |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
z |
|
B1u |
1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
-1 |
-1 |
1 |
|
|
B2u |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
Eu |
2 |
0 |
-2 |
0 |
0 |
-2 |
0 |
2 |
0 |
0 |
(x,y) |
|
D4d |
E |
2S8 |
2C4 |
2S83 |
2C2 |
4C2’ |
4sigmad |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2 |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1 |
1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
|
|
B2 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
z |
|
E1 |
2 |
√2 |
0 |
-√2 |
-2 |
0 |
0 |
(x,y) |
|
E2 |
2 |
0 |
-2 |
0 |
2 |
0 |
0 |
(x2-y2,xy) |
|
E3 |
2 |
-√2 |
0 |
√2 |
-2 |
0 |
0 |
(Rx,Ry), (xz,yz) |
|
S4=-4 |
E |
S4 |
C2 |
S43 |
|
|
A |
1 |
1 |
1 |
1 |
Rz, x2+y2, z2 |
|
B |
1 |
-1 |
1 |
-1 |
z, x2-y2, xy |
|
E |
{1,1} |
{i,-i} |
{-1,-1} |
{-i,i} |
(x,y), (Rx,Ry), (xz,yz) |
|
C6=6 |
E |
C6 |
C62 |
C63 |
C64 |
C65 |
|
|
A |
1 |
1 |
1 |
1 |
1 |
1 |
z, Rz,x2+y2, z2 |
|
B |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
E1 |
{1,1} |
{e, e*} |
{-e*, -e} |
{-1, -1} |
{-e, -e*} |
{e*,-e} |
(x,y), (Rx,Ry), (xz, yz) |
|
E2 |
{1,1} |
{-e*, -e} |
{-e, -e*} |
{1,1} |
{-e*, -e} |
{-e, -e*} |
(x2-y2,xy) |
|
C6h=6/m |
E |
C6 |
C62 |
C63 |
C64 |
C65 |
i |
S35 |
S65 |
sigmah |
S6 |
S3 |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
Rz, x2+y2, z2 |
|
Bg |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
E1g |
{1,1} |
{e,e*} |
{-e*,-e} |
{-1,-1} |
{-e,-e*} |
{e*,e} |
{1,1} |
{e,e*} |
{-e*,-e} |
{-1,-1} |
{-e,-e*} |
{e*,e} |
(Rx,Ry),(yz,zx) |
|
E2g |
{1,1} |
{-e*,-e} |
{-e,-e*} |
{1,1} |
{-e*,-e} |
{-e,-e*} |
{1,1} |
{-e*,-e} |
{-e,-e*} |
{1,1} |
{-e*,-e} |
{-e,-e*} |
(x2-y2,xy) |
|
Au |
1 |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
-1 |
z |
|
Bu |
1 |
-1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
|
|
E1u |
{1,1} |
{e,e*} |
{-e*,-e} |
{-1,-1} |
{-e,-e*} |
{e*,e} |
{-1,-1} |
{-e,-e*} |
{e*,e} |
{1,1} |
{e,e*} |
{-e*,-e} |
(x,y) |
|
E2u |
v |
{-e*,-e} |
{-e,-e*} |
{1,1} |
{-e*,-e} |
{-e,-e*} |
{-1,-1} |
{e*,e} |
{e,e*} |
{-1,-1} |
{e*,e} |
{e,e*} |
|
|
C6v=6mm |
E |
2C6 |
2C3 |
C2 |
3sigmah |
3sigmad |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
z, x2+y2, z2 |
|
A2 |
1 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
B2 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
|
|
E1 |
2 |
1 |
-1 |
-2 |
0 |
0 |
(x,y), (Rx,Ry), (xz, yz) |
|
E2 |
2 |
-1 |
-1 |
2 |
0 |
0 |
(x2-y2,xy) |
|
D6=622 |
E |
2C6 |
2C62 |
C63 |
3C2’ |
3C2’’ |
|
||||||||
|
A1 |
1 |
1 |
1 |
1 |
1 |
1 |
z, x2+y2, z2 |
|
|||||||
|
A2 |
1 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
|||||||
|
B1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
|||||||
|
B2 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
|
|
|||||||
|
E1 |
2 |
1 |
-1 |
-2 |
0 |
0 |
(x,y), (Rx,Ry), (xz, yz) |
|
|||||||
|
E2 |
2 |
-1 |
-1 |
2 |
0 |
0 |
(x2-y2,xy) |
|
|||||||
|
D6h=6/mmm |
E |
2C6 |
2C3 |
C2 |
3C2’ |
2C2’’ |
i |
2S2 |
2S6 |
sigmah |
3sigmad |
3sigmav |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2, z2 |
|
A2g |
1 |
1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
Rz |
|
B1g |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
-1 |
|
|
B2g |
1 |
-1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
|
|
E1g |
2 |
1 |
-1 |
-2 |
0 |
0 |
2 |
1 |
-1 |
-2 |
0 |
0 |
(Rx,Ry), (xz,yz) |
|
E2g |
2 |
-1 |
-1 |
2 |
0 |
0 |
2 |
-1 |
-1 |
2 |
0 |
0 |
(x2-y2,xy) |
|
A1u |
1 |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
-1 |
|
|
A2u |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
z |
|
B1u |
1 |
-1 |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
|
|
B2u |
1 |
-1 |
1 |
-1 |
-1 |
1 |
-1 |
1 |
-1 |
1 |
1 |
-1 |
|
|
E1u |
2 |
1 |
-1 |
-2 |
0 |
0 |
-2 |
-1 |
1 |
2 |
0 |
0 |
(x,y) |
|
E2u |
2 |
-1 |
-1 |
2 |
0 |
0 |
-2 |
1 |
1 |
-2 |
0 |
0 |
|
|
T=23 |
E |
4C3 |
4C32 |
3C2 |
|
|
A |
1 |
1 |
1 |
1 |
x2+y2+z2 |
|
E |
{1,1} |
{e,e*} |
{e*,e} |
{1,1} |
(x2-y2,2z2-x2-y2) |
|
T |
3 |
0 |
0 |
-1 |
(Rx,Ry,Rz),(x,y,z),(xy,xz,yz) |
|
Th=m3 |
E |
4C3 |
4C32 |
3C2 |
i |
4S6 |
4S65 |
sigmah |
|
|
Ag |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2+z2 |
|
Eg |
{1,1} |
{e,e*} |
{e*,e} |
{1,1} |
{1,1} |
{e,e*} |
{e*,e} |
{1,1} |
(x2-y2,2z2-x2-y2) |
|
Tg |
3 |
0 |
0 |
1 |
0 |
0 |
0 |
-1 |
(Rx,Ry,Rz),(xy,xz,yz) |
|
Au |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
|
|
Eu |
{1,1} |
{e,e*} |
{e*,e} |
{-e,-e*} |
{-1,-1} |
{-e,-e*} |
{-e*,-e} |
{-1,-1} |
|
|
Tu |
3 |
0 |
0 |
0 |
-1 |
0 |
0 |
1 |
(x,y,z) |
|
Td=-43m |
E |
8C3 |
3C2 |
6S43 |
6sigmad |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
x2+y2+z2 |
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
|
|
E |
2 |
-1 |
2 |
0 |
0 |
(x2-y2,2z2-x2-y2) |
|
T1 |
3 |
0 |
-1 |
1 |
-1 |
(Rx,Ry,Rz) |
|
T2 |
3 |
0 |
-1 |
-1 |
1 |
(x,y,z), (xy,xz,yz) |
|
O=432 |
E |
8C3 |
3C2 |
6C4 |
6C2 |
|
|
A1 |
1 |
1 |
1 |
1 |
1 |
x2+y2+z2 |
|
A2 |
1 |
1 |
1 |
-1 |
-1 |
|
|
E |
2 |
-1 |
2 |
0 |
0 |
(x2-y2,2z2-x2-y2) |
|
T1 |
3 |
0 |
-1 |
1 |
-1 |
(Rx,Ry,Rz) |
|
T2 |
3 |
0 |
-1 |
-1 |
1 |
(xy,xz,yz) |
|
Oh=m-3m |
E |
8C3 |
3C2 |
6C2’ |
6C4 |
i |
8S6 |
3sigmah |
6S4 |
6sigmad |
|
|
A1g |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
x2+y2+z2 |
|
A2g |
1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
1 |
-1 |
-1 |
|
|
Eg |
2 |
-1 |
2 |
0 |
0 |
2 |
-1 |
2 |
0 |
0 |
(x2-y2,2z2-x2-y2) |
|
T1g |
3 |
0 |
-1 |
1 |
-1 |
3 |
0 |
-1 |
1 |
-1 |
(Rx,Ry,Rz) |
|
T2g |
3 |
0 |
-1 |
-1 |
1 |
3 |
0 |
-1 |
-1 |
1 |
(xy,xz,yz) |
|
A1u |
1 |
1 |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
|
|
A2u |
1 |
1 |
1 |
-1 |
-1 |
-1 |
-1 |
-1 |
1 |
1 |
|
|
Eu |
2 |
-1 |
2 |
0 |
0 |
-2 |
1 |
-2 |
0 |
0 |
|
|
T1g |
3 |
0 |
-1 |
1 |
-1 |
-3 |
0 |
1 |
-1 |
1 |
(x,y,z) |
|
T2g |
3 |
0 |
-1 |
-1 |
1 |
-3 |
0 |
1 |
1 |
-1 |
|