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List of character tables for chemically important 3D point groups
List article

This lists the character tables for the more common molecular point groups used in the study of molecular symmetry. These tables are based on the group-theoretical treatment of the symmetry operations present in common molecules, and are useful in molecular spectroscopy and quantum chemistry. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.

Notation

For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Zn: cyclic group of order n, Dn: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, Sn: symmetric group on n letters, and An: alternating group on n letters.

The character tables then follow for all groups. The rows of the character tables correspond to the irreducible representations of the group, with their conventional names, known as Mulliken symbols,6 in the left margin. The naming conventions are as follows:

  • A and B are singly degenerate representations, with the former transforming symmetrically around the principal axis of the group, and the latter asymmetrically. E, T, G, H, ... are doubly, triply, quadruply, quintuply, ... degenerate representations.
  • g and u subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion. Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis. Higher numbers denote additional representations with such asymmetry.
  • Single prime ( ' ) and double prime ( '' ) superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal mirror plane σh, one perpendicular to the principal rotation axis.

All but the two rightmost columns correspond to the symmetry operations which are invariant in the group. In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.

The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations. The symbol i used in the body of the table denotes the imaginary unit: i 2 = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes complex conjugation.

The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (Rx, Ry and Rz), and functions of the quadratic terms of the coordinates(x2, y2, z2, xy, xz, and yz).

A further column is included in some tables, such as those of Salthouse and Ware7 For example,

C s {\displaystyle C_{s}} E {\displaystyle E} σ h {\displaystyle \sigma _{h}}
A ′ {\displaystyle A'} 1 {\displaystyle 1} 1 {\displaystyle 1} x {\displaystyle x} , y {\displaystyle y} , R z {\displaystyle R_{z}} x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , z 2 {\displaystyle z^{2}} , x y {\displaystyle xy} z x 2 {\displaystyle zx^{2}} , y z 2 {\displaystyle yz^{2}} , x 2 y {\displaystyle x^{2}y} , x y 2 {\displaystyle xy^{2}} , x 3 {\displaystyle x^{3}} , y 3 {\displaystyle y^{3}}
A ″ {\displaystyle A''} 1 {\displaystyle 1} − 1 {\displaystyle -1} z {\displaystyle z} , R x {\displaystyle R_{x}} , R y {\displaystyle R_{y}} y z {\displaystyle yz} , x z {\displaystyle xz} z 3 {\displaystyle z^{3}} , x y z {\displaystyle xyz} , y 2 z {\displaystyle y^{2}z} , x 2 z {\displaystyle x^{2}z}

The last column relates to cubic functions which may be used in applications regarding f orbitals in atoms.

Character tables

Nonaxial symmetries

These groups are characterized by a lack of a proper rotation axis, noting that a C 1 {\displaystyle C_{1}} rotation is considered the identity operation. These groups have involutional symmetry: the only nonidentity operation, if any, is its own inverse.

In the group C 1 {\displaystyle C_{1}} , all functions of the Cartesian coordinates and rotations about them transform as the A {\displaystyle A} irreducible representation.

Point GroupCanonical GroupOrderCharacter Table
C 1 {\displaystyle C_{1}} Z 1 {\displaystyle Z_{1}} 1 {\displaystyle 1}
E {\displaystyle E}
A {\displaystyle A} 1 {\displaystyle 1}
C i {\displaystyle C_{i}} Z 2 {\displaystyle Z_{2}} 2
E {\displaystyle E} i {\displaystyle i}
A g {\displaystyle A_{g}} 1 {\displaystyle 1} 1 {\displaystyle 1} R x {\displaystyle R_{x}} , R y {\displaystyle R_{y}} , R z {\displaystyle R_{z}} x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , z 2 {\displaystyle z^{2}} , x y {\displaystyle xy} , x z {\displaystyle xz} , y z {\displaystyle yz}
A u {\displaystyle A_{u}} 1 {\displaystyle 1} − 1 {\displaystyle -1} x {\displaystyle x} , y {\displaystyle y} , z {\displaystyle z}
C s {\displaystyle C_{s}} Z 2 {\displaystyle Z_{2}} 2 {\displaystyle 2}
E {\displaystyle E} σ h {\displaystyle \sigma _{h}}
A ′ {\displaystyle A'} 1 {\displaystyle 1} 1 {\displaystyle 1} x {\displaystyle x} , y {\displaystyle y} , R z {\displaystyle R_{z}} x 2 {\displaystyle x^{2}} , y 2 {\displaystyle y^{2}} , z 2 {\displaystyle z^{2}} , x y {\displaystyle xy}
A ″ {\displaystyle A''} 1 {\displaystyle 1} − 1 {\displaystyle -1} z {\displaystyle z} , R x {\displaystyle R_{x}} , R y {\displaystyle R_{y}} y z {\displaystyle yz} , x z {\displaystyle xz}

Cyclic symmetries

The families of groups with these symmetries have only one rotation axis.

Cyclic groups (Cn)

The cyclic groups are denoted by Cn. These groups are characterized by an n-fold proper rotation axis Cn. The C1 group is covered in the nonaxial groups section.

PointGroupCanonicalGroupOrderCharacter Table
C2Z22
 EC2  
A11Rz, zx2, y2, z2, xy
B1−1Rx, Ry, x, yxz, yz
C3Z33
 EC3 C32θ = e2πi /3
A111Rz, zx2 + y2
E1 1θ  θCθC θ (Rx, Ry), (x, y)(x2 - y2, xy), (xz, yz)
C4Z44
 EC4 C2 C43 
A1111Rz, zx2 + y2, z2
B1−11−1 x2 − y2, xy
E1 1i −i−1 −1−i i(Rx, Ry), (x, y)(xz, yz)
C5Z55
 E  C5 C52C53C54θ = e2πi /5
A11111Rz, zx2 + y2, z2
E11 1θ  θCθ2 (θ2)C(θ2)C θ2θC θ (Rx, Ry), (x, y)(xz, yz)
E21 1θ2 (θ2)CθC θ θ  θC(θ2)C θ2 (x2 - y2, xy)
C6Z66
 E  C6 C3 C2 C32C65θ = e2πi /6
A111111Rz, zx2 + y2, z2
B1−11−11−1  
E11 1θ  θC−θC −θ −1 −1−θ  −θCθC −θ (Rx, Ry), (x, y)(xz, yz)
E21 1−θC −θ −θ  −θC1 1−θC −θ −θ  −θC (x2 − y2, xy)
C8Z88
 E  C8 C4 C83C2 C85C43C87θ = e2πi /8
A11111111Rz, zx2 + y2, z2
B1−11−11−11−1  
E11 1θ  θCi −i−θC −θ −1 −1−θ  −θC−i iθC θ (Rx, Ry), (x, y)(xz, yz)
E21 1i −i−1 −1−i i1 1i −i−1 −1−i i (x2 − y2, xy)
E31 1−θ  −θCi −iθC θ −1 −1θ  θC−i i−θC −θ   

Reflection groups (Cnh)

The reflection groups are denoted by Cnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) a mirror plane σh normal to Cn. The C1h group is the same as the Cs group in the nonaxial groups section.

PointGroupCanonicalgroupOrderCharacter Table
C2hZ2 × Z24
 EC2 iσh  
Ag1111Rzx2, y2, z2, xy
Bg1−11−1Rx, Ryxz, yz
Au11−1−1z 
Bu1−1−11x, y 
C3hZ66
 EC3 C32σh S3 S35θ = e2πi /3
A'111111Rzx2 + y2, z2
E'1 1θ  θCθC θ 1 1θ  θCθC θ (x, y)(x2 − y2, xy)
A''111−1−1−1z 
E''1 1θ  θCθC θ −1 −1−θ  −θC−θC −θ (Rx, Ry)(xz, yz)
C4hZ2 × Z48
 EC4 C2 C43iS43σh S4  
Ag11111111Rzx2 + y2, z2
Bg1−11−11−11−1 x2 − y2, xy
Eg1 1i −i−1 −1−i i1 1i −i−1 −1−i i(Rx, Ry)(xz, yz)
Au1111−1−1−1−1z 
Bu1−11−1−11−11  
Eu1 1i −i−1 −1−i i−1 −1−i i1 1i −i(x, y) 
C5hZ1010
 E  C5 C52C53C54σh S5 S57S53S59θ = e2πi /5
A'1111111111Rzx2 + y2, z2
E1'1 1θ  θCθ2 (θ2)C(θ2)C θ2θC θ 1 1θ  θCθ2 (θ2)C(θ2)C θ2θC θ (x, y) 
E2'1 1θ2 (θ2)CθC θ θ  θC(θ2)C θ21 1θ2 (θ2)CθC θ θ  θC(θ2)C θ2 (x2 - y2, xy)
A''11111−1−1−1−1−1z 
E1''1 1θ  θCθ2 (θ2)C(θ2)C θ2θC θ −1 −1−θ  -θC−θ2 −(θ2)C−(θ2)C −θ2−θC −θ (Rx, Ry)(xz, yz)
E2''1 1θ2 (θ2)CθC θ θ  θC(θ2)C θ2−1 −1−θ2 −(θ2)C−θC −θ −θ  −θC−(θ2)C −θ2  
C6hZ2 × Z612
 E  C6 C3 C2 C32C65iS35S65σh S6 S3 θ = e2πi /6
Ag111111111111Rzx2 + y2, z2
Bg1−11−11−11−11−11−1  
E1g1 1θ  θC−θC −θ −1 −1−θ  −θCθC θ 1 1θ  θC−θC −θ −1 −1−θ  −θCθC θ (Rx, Ry)(xz, yz)
E2g1 1−θC −θ −θ  −θC1 1−θC −θ −θ  −θC1 1−θC −θ −θ  −θC1 1−θC −θ −θ  −θC (x2 − y2, xy)
Au111111−1−1−1−1−1−1z 
Bu1−11−11−1−11−11−11  
E1u1 1θ  θC−θC −θ −1 −1−θ  −θCθC θ −1 −1−θ  −θCθC θ 1 1θ  θC−θC −θ (x, y) 
E2u1 1−θC −θ −θ  −θC1 1−θC −θ −θ  −θC−1 −1θC θ θ  θC−1 −1θC θ θ  θC  

Pyramidal groups (Cnv)

The pyramidal groups are denoted by Cnv. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n mirror planes σv which contain Cn. The C1v group is the same as the Cs group in the nonaxial groups section.

PointGroupCanonicalgroupOrderCharacter Table
C2vZ2 × Z2 (=D2)4
 EC2 σv σv'  
A11111zx2 , y2, z2
A211−1−1Rzxy
B11−11−1Ry, xxz
B21−1−11Rx, yyz
C3vD36
 E2 C3 3 σv  
A1111zx2 + y2, z2
A211−1Rz 
E2−10(Rx, Ry), (x, y)(x2 − y2, xy), (xz, yz)
C4vD48
 E2 C4 C2 2 σv 2 σd  
A111111zx2 + y2, z2
A2111−1−1Rz 
B11−111−1 x2 − y2
B21−11−11 xy
E20−200(Rx, Ry), (x, y)(xz, yz)
C5vD510
 E  2 C5 2 C525 σv θ = 2π/5
A11111zx2 + y2, z2
A2111−1Rz 
E122 cos(θ)2 cos(2θ)0(Rx, Ry), (x, y)(xz, yz)
E222 cos(2θ)2 cos(θ)0 (x2 − y2, xy)
C6vD612
 E  2 C6 2 C3 C2 3 σv 3 σd  
A1111111zx2 + y2, z2
A21111−1−1Rz 
B11−11−11−1  
B21−11−1−11  
E121−1−200(Rx, Ry), (x, y)(xz, yz)
E22−1−1200 (x2 − y2, xy)

Improper rotation groups (Sn)

The improper rotation groups are denoted by Sn. These groups are characterized by an n-fold improper rotation axis Sn, where n is necessarily even. The S2 group is the same as the Ci group in the nonaxial groups section. Sn groups with an odd value of n are identical to Cnh groups of same n and are therefore not considered here (in particular, S1 is identical to Cs).

The S8 table reflects the 2007 discovery of errors in older references.8 Specifically, (Rx, Ry) transform not as E1 but rather as E3.

PointGroupCanonicalgroupOrderCharacter Table
S4Z44
 ES4 C2 S43 
A1111Rz,  x2 + y2, z2
B1−11−1zx2 − y2, xy
E1 1i −i−1 −1−i i(Rx, Ry), (x, y)(xz, yz)
S6Z66
 E  S6 C3 iC32S65θ = e2πi /6
Ag111111Rzx2 + y2, z2
Eg1 1θC θ θ  θC1 1θC θ θ  θC(Rx, Ry)(x2 − y2, xy), (xz, yz)
Au1−11−11−1z 
Eu1 1−θC −θ θ  θC−1 −1θC θ −θ  −θC(x, y) 
S8Z88
 E  S8 C4 S83iS85C42S87θ = e2πi /8
A11111111Rzx2 + y2, z2
B1−11−1−1−11−1z 
E11 1θ  θCi −i−θC −θ −1 −1−θ  −θC−i iθC θ (x, y)(xz, yz)
E21 1i −i−1 −1−i i1 1i −i−1 −1−i i (x2 − y2, xy)
E31 1−θC −θ −i iθ  θC−1 −1θC θ i −i−θ −θC(Rx, Ry)(xz, yz)

Dihedral symmetries

The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.

Dihedral groups (Dn)

The dihedral groups are denoted by Dn. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn. The D1 group is the same as the C2 group in the cyclic groups section.

PointGroupCanonicalgroupOrderCharacter Table
D2Z2 × Z2(=D2)4
 EC2 (z)C2 (x)C2 (y) 
A1111 x2, y2, z2
B111−1−1Rz, zxy
B21−1−11Ry, yxz
B31−11−1Rx, xyz
D3D36
 E2 C3 3 C'2  
A1111 x2 + y2, z2
A211−1Rz, z 
E2−10(Rx, Ry), (x, y)(x2 − y2, xy), (xz, yz)
D4D48
 E2 C4 C2 2 C2' 2 C2''  
A111111 x2 + y2, z2
A2111−1−1Rz, z 
B11−111−1 x2 − y2
B21−11−11 xy
E20−200(Rx, Ry), (x, y)(xz, yz)
D5D510
 E  2 C5 2 C525 C2 θ=2π/5
A11111 x2 + y2, z2
A2111−1Rz, z 
E122 cos(θ)2 cos(2θ)0(Rx, Ry), (x, y)(xz, yz)
E222 cos(2θ)2 cos(θ)0 (x2 − y2, xy)
D6D612
 E  2 C6 2 C3 C2 3 C2' 3 C2''  
A1111111 x2 + y2, z2
A21111−1−1Rz, z 
B11−11−11−1  
B21−11−1−11  
E121−1−200(Rx, Ry), (x, y)(xz, yz)
E22−1−1200 (x2 − y2, xy)

Prismatic groups (Dnh)

The prismatic groups are denoted by Dnh. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) a mirror plane σh normal to Cn and containing the C2s. The D1h group is the same as the C2v group in the pyramidal groups section.

The D8h table reflects the 2007 discovery of errors in older references.9 Specifically, symmetry operation column headers 2S8 and 2S83 were reversed in the older references.

PointGroupCanonicalgroupOrderCharacter Table
D2hZ2×Z2×Z2(=Z2×D2)8
 EC2 C2 (x)C2 (y)iσ(xy)  σ(xz)  σ(yz)   
Ag11111111 x2, y2, z2
B1g11−1−111−1−1Rzxy
B2g1−1−111−11−1Ryxz
B3g1−11−11−1−11Rxyz
Au1111−1−1−1−1  
B1u11−1−1−1−111z 
B2u1−1−11−11−11y 
B3u1−11−1−111−1x 
D3hD612
 E2 C3 3 C2 σh 2 S3 3 σv  
A1'111111 x2 + y2, z2
A2'11−111−1Rz 
E'2−102−10(x, y)(x2 − y2, xy)
A1''111−1−1−1  
A2''11−1−1−11z 
E''2−10−210(Rx, Ry)(xz, yz)
D4hZ2×D416
 E2 C4 C2 2 C2' 2 C2'' i2 S4 σh 2 σv 2 σd  
A1g1111111111 x2 + y2, z2
A2g111−1−1111−1−1Rz 
B1g1−111−11−111−1 x2 − y2
B2g1−11−111−11−11 xy
Eg20−20020−200(Rx, Ry)(xz, yz)
A1u11111−1−1−1−1−1  
A2u111−1−1−1−1−111z 
B1u1−111−1−11−1−11  
B2u1−11−11−11−11−1  
Eu20−200−20200(x, y) 
D5hD1020
 E  2 C5 2 C525 C2 σh 2 S5 2 S535 σv θ=2π/5
A1'11111111 x2 + y2, z2
A2'111−1111−1Rz 
E1'22 cos(θ)2 cos(2θ)022 cos(θ)2 cos(2θ)0(x, y) 
E2'22 cos(2θ)2 cos(θ)022 cos(2θ)2 cos(θ)0 (x2 − y2, xy)
A1''1111−1−1−1−1  
A2''111−1−1−1−11z 
E1''22 cos(θ)2 cos(2θ)0−2−2 cos(θ)−2 cos(2θ)0(Rx, Ry)(xz, yz)
E2''22 cos(2θ)2 cos(θ)0−2−2 cos(2θ)−2 cos(θ)0  
D6hZ2×D624
 E  2 C6 2 C3 C2 3 C2' 3 C2'' i2 S3 2 S6 σh 3 σd 3 σv  
A1g111111111111 x2 + y2, z2
A2g1111−1−11111−1−1Rz 
B1g1−11−11−11−11−11−1  
B2g1−11−1−111−11−1−11  
E1g21−1−20021−1−200(Rx, Ry)(xz, yz)
E2g2−1−12002−1−1200 (x2 − y2, xy)
A1u111111−1−1−1−1−1−1  
A2u1111−1−1−1−1−1−111z 
B1u1−11−11−1−11−11−11  
B2u1−11−1−11−11−111−1  
E1u21−1−200−2−11200(x, y) 
E2u2−1−1200−211−200  
D8hZ2×D832
 E  2 C8 2 C832 C4 C2 4 C2' 4 C2'' i2 S832 S8 2 S4 σh 4 σd 4 σv θ=21/2
A1g11111111111111 x2 + y2, z2
A2g11111−1−111111−1−1Rz 
B1g1−1−1111−11−1−1111−1  
B2g1−1−111−111−1−111−11  
E1g2θ−θ0−2002θ−θ0−200(Rx, Ry)(xz, yz)
E2g200−2200200−2200 (x2 − y2, xy)
E3g2−θθ0−2002−θθ0−200  
A1u1111111−1−1−1−1−1−1−1  
A2u11111−1−1−1−1−1−1−111z 
B1u1−1−1111−1−111−1−1−11  
B2u1−1−111−11−111−1−11−1  
E1u2θ−θ0−200−2−θθ0200(x, y) 
E2u200−2200−2002−200  
E3u2−θθ0−200−2θ−θ0200  

Antiprismatic groups (Dnd)

The antiprismatic groups are denoted by Dnd. These groups are characterized by i) an n-fold proper rotation axis Cn; ii) n 2-fold proper rotation axes C2 normal to Cn; iii) n mirror planes σd which contain Cn. The D1d group is the same as the C2h group in the reflection groups section.

PointGroupCanonicalgroupOrderCharacter Table
D2dD48
 E 2 S4 C2 2 C2' 2 σd  
A111111 x2, y2, z2
A2111−1−1Rz 
B11−111−1 x2 − y2
B21−11−11zxy
E20−200(Rx, Ry), (x, y)(xz, yz)
D3dD612
 E 2 C3 3 C2 i 2 S6 3 σd  
A1g111111 x2 + y2, z2
A2g11−111−1Rz 
Eg2−102−10(Rx, Ry)(x2 − y2, xy), (xz, yz)
A1u111−1−1−1  
A2u11−1−1−11z 
Eu2−10−210(x, y) 
D4dD816
 E 2 S8 2 C4 2 S83C2 4 C2' 4 σd θ=21/2
A11111111 x2 + y2, z2
A211111−1−1Rz 
B11−11−111−1  
B21−11−11−11z 
E12θ0−θ−200(x, y) 
E220−20200 (x2 − y2, xy)
E32−θ0θ−200(Rx, Ry)(xz, yz)
D5dD1020
 E  2 C5 2 C525 C2 i 2 S10 2 S1035 σd θ=2π/5
A1g11111111 x2 + y2, z2
A2g111−1111−1Rz 
E1g22 cos(θ)2 cos(2θ)022 cos(2θ)2 cos(θ)0(Rx, Ry)(xz, yz)
E2g22 cos(2θ)2 cos(θ)022 cos(θ)2 cos(2θ)0 (x2 − y2, xy)
A1u1111−1−1−1−1  
A2u111−1−1−1−11z 
E1u22 cos(θ)2 cos(2θ)0−2−2 cos(2θ)−2 cos(θ)0(x, y) 
E2u22 cos(2θ)2 cos(θ)0−2−2 cos(θ)−2 cos(2θ)0  
D6dD1224
 E  2 S12 2 C6 2 S4 2 C3 2 S125C2 6 C2' 6 σd θ=31/2
A1111111111 x2 + y2, z2
A21111111−1−1Rz 
B11−11−11−111−1  
B21−11−11−11−11z 
E12θ10−1−θ−200(x, y) 
E221−1−2−11200 (x2 − y2, xy)
E320−2020−200  
E42−1−12−1−1200  
E52−θ10−1θ−200(Rx, Ry)(xz, yz)

Polyhedral symmetries

These symmetries are characterized by having more than one proper rotation axis of order greater than 2.

Cubic groups

These polyhedral groups are characterized by not having a C5 proper rotation axis.

PointGroupCanonicalgroupOrderCharacter Table
TA412
 E4 C3 4 C323 C2 θ=e2π i/3
A1111 x2 + y2 + z2
E1 1θ  θCθC θ 1 1 (2 z2 − x2 − y2, x2 − y2)
T300−1(Rx, Ry, Rz),(x, y, z)(xy, xz, yz)
TdS424
 E8 C3 3 C2 6 S4 6 σd  
A111111 x2 + y2 + z2
A2111−1−1  
E2−1200 (2 z2 − x2 − y2, x2 − y2)
T130−11−1(Rx, Ry, Rz) 
T230−1−11(x, y, z)(xy, xz, yz)
ThZ2×A424
 E4 C3 4 C323 C2 i4 S6 4 S653 σh θ=e2π i/3
Ag11111111 x2 + y2 + z2
Au1111−1−1−1−1  
Eg1 1θ  θCθC θ 1 11 1θ  θCθC θ 1 1 (2 z2 − x2 − y2, x2 − y2)
Eu1 1θ  θCθC θ 1 1−1 −1−θ  −θC−θC −θ −1 −1  
Tg300−1300−1(Rx, Ry, Rz)(xy, xz, yz)
Tu300−1−3001(x, y, z) 
OS424
 E  6 C4 3 C2  (C42)8 C3 6 C'2  
A111111 x2 + y2 + z2
A21−111−1  
E202−10 (2 z2 − x2 − y2, x2 − y2)
T131−10−1(Rx, Ry, Rz), (x, y, z) 
T23−1−101 (xy, xz, yz)
OhZ2×S448
 E  8 C3 6 C2 6 C4 3 C2  (C42)i6 S4 8 S6 3 σh 6 σd  
A1g1111111111 x2 + y2 + z2
A2g11−1−111−111−1  
Eg2−100220−120 (2 z2 − x2 − y2, x2 − y2)
T1g30−11−1310−1−1(Rx, Ry, Rz) 
T2g301−1−13−10−11 (xy, xz, yz)
A1u11111−1−1−1−1−1  
A2u11−1−11−11−1−11  
Eu2−1002−201−20  
T1u30−11−1−3−1011(x, y, z) 
T2u301−1−1−3101−1  

Icosahedral groups

See also: Icosahedral symmetry

These polyhedral groups are characterized by having a C5 proper rotation axis.

PointGroupCanonicalgroupOrderCharacter Table
IA560
 E12 C5 12 C5220 C3 15 C2 θ=π/5
A11111 x2 + y2 + z2
T132 cos(θ)2 cos(3θ)0−1(Rx, Ry, Rz),(x, y, z) 
T232 cos(3θ)2 cos(θ)0−1  
G4−1−110  
H500−11 (2 z2 − x2 − y2, x2 − y2, xy, xz, yz)
IhZ2×A5120
 E12 C5 12 C5220 C3 15 C2 i12 S10 12 S10320 S6 15 σθ=π/5
Ag1111111111 x2 + y2 + z2
T1g32 cos(θ)2 cos(3θ)0−132 cos(3θ)2 cos(θ)0−1(Rx, Ry, Rz) 
T2g32 cos(3θ)2 cos(θ)0−132 cos(θ)2 cos(3θ)0−1  
Gg4−1−1104−1−110  
Hg500−11500−11 (2 z2 − x2 − y2, x2 − y2, xy, xz, yz)
Au11111−1−1−1−1−1  
T1u32 cos(θ)2 cos(3θ)0−1−3−2 cos(3θ)−2 cos(θ)01(x, y, z) 
T2u32 cos(3θ)2 cos(θ)0−1−3−2 cos(θ)−2 cos(3θ)01  
Gu4−1−110−411−10  
Hu500−11−5001−1  

Linear (cylindrical) groups

These groups are characterized by having a proper rotation axis C∞ around which the symmetry is invariant to any rotation.

PointGroupCharacter Table
C∞v
 E2 C∞Φ...∞ σv  
A1=Σ+11...1zx2 + y2, z2
A2=Σ−11...−1Rz 
E1=Π22 cos(Φ)...0(x, y), (Rx, Ry)(xz, yz)
E2=Δ22 cos(2Φ)...0 (x2 - y2, xy)
E3=Φ22 cos(3Φ)...0  
...............  
D∞h
 E2 C∞Φ...∞ σv i2 S∞Φ...∞ C2  
Σg+11...111...1 x2 + y2, z2
Σg−11...−111...−1Rz 
Πg22 cos(Φ)...02−2 cos(Φ)..0(Rx, Ry)(xz, yz)
Δg22 cos(2Φ)...022 cos(2Φ)..0 (x2 − y2, xy)
...........................  
Σu+11...1−1−1...−1z 
Σu−11...−1−1−1...1  
Πu22 cos(Φ)...0−22 cos(Φ)..0(x, y) 
Δu22 cos(2Φ)...0−2−2 cos(2Φ)..0  
...........................  

See also

Notes

Further reading

  • Bunker, Philip; Jensen, Per (2006). Molecular Symmetry and Spectroscopy, Second edition. Ottawa: NRC Research Press. ISBN 0-660-19628-X.

References

  1. Drago, Russell S. (1977). Physical Methods in Chemistry. W.B. Saunders Company. ISBN 0-7216-3184-3. 0-7216-3184-3 ↩

  2. Cotton, F. Albert (1990). Chemical Applications of Group Theory. John Wiley & Sons: New York. ISBN 0-471-51094-7. 0-471-51094-7 ↩

  3. Gelessus, Achim (2007-07-12). "Character tables for chemically important point groups". Jacobs University, Bremin; Computational Laboratory for Analysis, Modeling, and Visualization. Retrieved 2007-07-12. http://symmetry.jacobs-university.de/ ↩

  4. Shirts, Randall B. (2007). "Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables". Journal of Chemical Education. 84 (1882). American Chemical Society: 1882. Bibcode:2007JChEd..84.1882S. doi:10.1021/ed084p1882. Retrieved 2007-10-16. http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html ↩

  5. Vanovschi, Vitalii. "POINT GROUP SYMMETRY CHARACTER TABLES". WebQC.Org. Retrieved 2008-10-29. http://www.webqc.org/symmetry.php ↩

  6. Mulliken, Robert S. (1933-02-15). "Electronic Structures of Polyatomic Molecules and Valence. IV. Electronic States, Quantum Theory of the Double Bond". Physical Review. 43 (4). American Physical Society (APS): 279–302. Bibcode:1933PhRv...43..279M. doi:10.1103/physrev.43.279. ISSN 0031-899X. /wiki/Robert_S._Mulliken ↩

  7. Salthouse, J.A.; Ware, M.J. (1972). Point group character tables and related data. Cambridge: Cambridge University Press. pp. 88 + v. ISBN 0-521-08139-4. 0-521-08139-4 ↩

  8. Shirts, Randall B. (2007). "Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables". Journal of Chemical Education. 84 (1882). American Chemical Society: 1882. Bibcode:2007JChEd..84.1882S. doi:10.1021/ed084p1882. Retrieved 2007-10-16. http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html ↩

  9. Shirts, Randall B. (2007). "Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables". Journal of Chemical Education. 84 (1882). American Chemical Society: 1882. Bibcode:2007JChEd..84.1882S. doi:10.1021/ed084p1882. Retrieved 2007-10-16. http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html ↩