Character Tables for the Crystal Point Groups Dr A.T. Boothroyd Notes The point groups are labelled by their international symbols (with the Schoenflies symbols in brackets). The classes of group elements are indicated by one typical element followed in square brackets by the number of elements in that class. The symbol n stands for a rotation by 2π/n, and nr for 2π r/n. The rotation axis is given by a suffix, e.g. x, y, and z; d means an axis at π/4 to Ox and Oy. The symbol mz means a reflection in a plane perpendicular to the z-axis. n– is an improper rotation by 2π/n, i.e. a proper rotation by this angle followed by inversion through the origin. An inversion on its own is – thus denoted by 1 . The irreducible representations are labelled according to two schemes, the first being the n notation of Bethe, following the convention of Koster, Dimmock, Wheeler and Statz in Properties of the Thirty-Two Point Groups, and the second is a systematic approach described as follows. A and B refer to one-dimensional representations, B being used if a rotation by 2π/n about the principal axis (chosen as the z-axis) has the character –1. E is used for a two-dimensional representation, and T for a threedimensional one. A pair of complex conjugate one-dimensional representations are always bracketed together and regarded as a two-dimensional representation E because time-reversal symmetry makes them degenerate. If there are two representations in which the characters of mz differ in sign, then they are distinguished by „ and “. Subscripts g and u (German: gerade and ungerade) indicate even and odd representations under inversion. In the event of both („,”) and (g,u) labellings being applicable, (g,u) takes precedence over („,”), which in turn precedes numerical subscripts, 1, 2, etc. –
Some point groups are direct products, G × 1 , where G contains only proper – – rotations R1, R2, etc, and 1 is the inversion group (E, 1 ). Their character tables may be constructed as follows. If D is an irreducible representation of G, then the group G × 1– will contain two corresponding irreducible representations, Dg and Du, with characters of opposite signs for the improper rotations, and the same sign for the proper rotations:
G
R D
G × 1–
R
– R
+
Dg
(R)
–
Du
(R) – (R)
(R)
(R)
On the right side of each character table are listed, in the row of the irreducible representation according to which they transform, the coordinate variables x, y, z, various products of these variables, and the infinitesimal rotations Rx, Ry and Rz (which transform as a pseudovector). For the direct product groups, x, y, z transform according to the appropriate ungerade representation, Du (or –), and x2, y2, z2, xy, yz, xz, Rx, Ry, Rz transform according to Dg ( +). Character Tables for the Crystal Point Groups
page 1
Triclinic, monoclinic and orthorhombic point groups
1 (C1)
E x, y, z, x2, y2, z2, xy, yz, xz, Rx, Ry, Rz
A
1
1 (S2/Ci)
E
1
1
–
1 1
–
+
Ag
1
1
–
Au
1
–1
E
2z
2 (C2)
x2, y2, z2, xy, yz, xz, Rx, Ry, Rz x, y, z
1
A
1
1
z, x2, y2, z2, xy, Rz
2
B
1
–1
x, y, yz, xz, Rx, Ry
E
mz
m (C1h/Cs)
x, y, x2, y2, z2, xy, Rz
1
A‟
1
1
2
A”
1
–1
z, yz, xz, Rx, Ry
E
2x
2y
2z
222 (D2/V) 1
A
1
1
1
1
x2, y2, z2
3
B1
1
–1
–1
1
z, xy, Rz
2
B2
1
–1
1
–1
y, xz, Ry
4
B3
1
1
–1
–1
x, yz, Rx
–
mmm (D2h/Vh) = 222 × 1
Character Tables for the Crystal Point Groups
page 2
2mm (C2v)
E
2z
my
mx
1
A1
1
1
1
1
3
A2
1
1
–1
–1
xy, Rz
2
B1
1
–1
1
–1
x, xz, Ry
4
B2
1
–1
–1
1
y, yz, Rx
2/m (C2h)
E
2z
mz
+
Ag
1
1
1
1
x2, y2, z2, xy, Rz
+
Bg
1
–1
–1
1
yz, xz, Rx, Ry
–
Au
1
1
–1
–1
z
–
Bu
1
–1
1
–1
x, y
E
2z
4z
1 2 1 2
z, x2, y2, z2
–
1
Tetragonal point groups
4 (C4)
(4z)3
1
A
1
1
1
1
2
B
1
1
–1
–1
1
–1
i
–i
1
–1
–i
i
3 4
E
{
z, x2+y2, z2, Rz x2–y2, xy (x, y), (xz, yz), (Rx, Ry)
–
4/m (C4h) = 4 × 1
–
4 (S4)
–
–
E
2z
4z
(4 z)3
1
A
1
1
1
1
x2+y2, z2, Rz
2
B
1
1
–1
–1
z, x2–y2, xy
1
–1
i
–i
1
–1
–i
i
3 4
E
{
Character Tables for the Crystal Point Groups
(x, y), (xz, yz), (Rx, Ry)
page 3
422 (D4)
E
2z
4z [2] 2x [2] 2d [2] x2+y2, z2
1
A1
1
1
1
1
1
2
A2
1
1
1
–1
–1
z, Rz
3
B1
1
1
–1
1
–1
x2–y2
4
B2
1
1
–1
–1
1
xy
5
E
2
–2
0
0
0
(x, y), (xz, yz), (Rx, Ry)
–
4/mmm (D4h) = 422 × 1
4mm (C4v)
E
2z
1
A1
1
1
1
1
1
2
A2
1
1
1
–1
–1
Rz
3
B1
1
1
–1
1
–1
x2–y2
4
B2
1
1
–1
–1
1
xy
5
E
2
–2
0
0
0
(x, y), (xz, yz), (Rx, Ry)
E
2z
–
4 2m (D2d/Vd)
4z [2] mx [2] md [2] z, x2+y2, z2
–
4 z [2] 2x [2] md [2] x2+y2, z2
1
A1
1
1
1
1
1
2
A2
1
1
1
–1
–1
Rz
3
B1
1
1
–1
1
–1
x2–y2
4
B2
1
1
–1
–1
1
z, xy
5
E
2
–2
0
0
0
(x, y), (xz, yz), (Rx, Ry)
Character Tables for the Crystal Point Groups
page 4
Trigonal and hexagonal point groups
3 (C3) A
1 2
E
3
{
E
3z
(3z)2
1
1
1
= e2πi/3 z, x2+y2, z2, Rz
2
1
(x, y), (x2–y2, xy), (yz, xz), (Rx, Ry)
2
1
–
–
3 (S6/C3i) = 3 × 1
32 (D3)
E
3z [2] 2y [3]
1
A1
1
1
1
2
A2
1
1
–1
3
E
2
–1
0
–
x2+y2, z2 z, Rz (x, y), (x2–y2, xy), (yz, xz), (Rx, Ry)
–
3 m (D3d) = 32 × 1
3m (C3v)
E
3z [2] mx [3]
1
A1
1
1
1
2
A2
1
1
–1
3
E
2
–1
0
Character Tables for the Crystal Point Groups
z, x2+y2, z2 Rz (x, y), (x2–y2, xy), (yz, xz), (Rx, Ry)
page 5
6 (C6)
E
6z
3z
2z
1
A
1
1
1
1
1
1
4
B
1
–1
1
–1
1
–1
–
1
–
5 6 3 2
E1
{
1
E2
{
1
2
–1
2
1
2
1
2
–1 2
= e2πi/3
(3z)2 (6z)5
z, x2+y2, z2, Rz
– 2
–
2 2
1
(x, y), (xz, yz), (Rx, Ry) x2–y2, xy
–
6/m (C6h) = 6 × 1 –
6 (C3h)
–
–
E
3z
(3z)2
mz
6z
(6 z)5
1
A‟
1
1
1
1
1
1
4
A”
1
1
1
–1
–1
–1
2
1
2 3 5
E‟
{
1
E”
{
1
6
622 (D6)
2
1 1
2
E
2z
x2+y2, z2, Rz z
2
(x, y), (x2–y2,xy)
2
1 2
= e2πi/3
–1
–
–1
–
2
– 2
–
(xz, yz), (Rx,Ry)
3z [2] 6z [2] 2y [3] 2x [3] x2+y2, z2
1
A1
1
1
1
1
1
1
2
A2
1
1
1
1
–1
–1
3
B1
1
–1
1
–1
1
–1
4
B2
1
–1
1
–1
–1
1
5
E1
2
–2
–1
1
0
0
(x, y), (xz, yz), (Rx, Ry)
6
E2
2
2
–1
–1
0
0
(x2–y2, xy)
z, Rz
–
6/mmm (D6h) = 622 × 1
Character Tables for the Crystal Point Groups
page 6
6mm (C6v)
E
2z
3z [2]
1
A1
1
1
1
1
1
1
2
A2
1
1
1
1
–1
–1
3
B1
1
–1
1
–1
–1
1
4
B2
1
–1
1
–1
1
–1
5
E1
2
–2
–1
1
0
0
(x, y), (xz, yz), (Rx, Ry)
6
E2
2
2
–1
–1
0
0
(x2–y2, xy)
6 m2 (D3h)
E
mz
1
A‟1
1
1
1
1
1
1
2
A‟2
1
1
1
1
–1
–1
3
A”1
1
–1
1
–1
1
–1
4
A”2
1
–1
1
–1
–1
1
z
6
E‟
2
2
–1
–1
0
0
(x, y), (x2–y2, xy)
5
E”
2
–2
–1
1
0
0
(xz, yz), (Rx, Ry)
–
Character Tables for the Crystal Point Groups
6z [2] my [3] mx [3] z, x2+y2, z2 Rz
–
3z [2] 6 z [2] 2y [3] mx [3] x2+y2, z2 Rz
page 7
Cubic point groups
23 (T)
E A
1 2
E
3
{
T
4
= e2πi/3
2z [3] 3 [4] (3)2 [4]
1
1
1
1
1
1
1
2
3
–1
0
1
x2+y2+z2 = r2
2
(x2–y2, 3z2–r2) 0
(x, y, z), (xy, yz, xz), (Rx, Ry, Rz)
–
m3 (Th) = 23 × 1
432 (O)
E
3 [8]
2z [3] 2d [6] 4z [6] x2+y2+z2 = r2
1
A1
1
1
1
1
1
2
A2
1
1
1
–1
–1
3
E
2
–1
2
0
0
(x2–y2, 3z2–r2)
4
T1
3
0
–1
–1
1
(x, y, z), (Rx, Ry, Rz)
5
T2
3
0
–1
1
–1
(xy, xz, yz)
–
m3m (Oh) = 432× 1
–
4 3m (Td)
E
3 [8]
–
2z [3] md [6] 4 z [6] x2+y2+z2 = r2
1
A1
1
1
1
1
1
2
A2
1
1
1
–1
–1
3
E
2
–1
2
0
0
(x2–y2, 3z2–r2)
4
T1
3
0
–1
–1
1
(Rx, Ry, Rz)
5
T2
3
0
–1
1
–1
Character Tables for the Crystal Point Groups
(x, y, z), (xy, xz, yz)
page 8
Character Tables for the Crystal Point Groups
page 9
