Character table for point group C4h

C4h E C4(z) C2 (C4)3 i (S4)3 h S4
linear functions,
rotations
quadratic
functions
cubic
functions
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+iRy
Rx-iRy
(xz, yz) -
Au +1 +1 +1 +1 -1 -1 -1 -1 z - z3, z(x2+y2)
Bu +1 -1 +1 -1 -1 +1 -1 +1 - - xyz, z(x2-y2)
Eu +1
+1
+i
-i
-1
-1
-i
+i
-1
-1
-i
+i
+1
+1
+i
-i
x+iy
x-iy
- (xz2, yz2) (xy2, x2y) (x3, y3)


Additional information

Number of symmetry elements h = 8
Number of irreducible representations n = 8
Number of real irreducible representations n = 6
Abelian group yes
Subgroups Cs , Ci , C2 , C4 , C2h , S4
Optical Isomerism (Chirality) no


Reduction formula for point group C4h

Type of representation

Information for point groups with complex irreducible representations

general 3N vib

E C4(z) C2 (C4)3 i (S4)3 h S4




Multipoles

dipole (p) Au+Eu
quadrupole (d) Ag+2Bg+Eg
octopole (f) Au+2Bu+2Eu
hexadecapole (g) 3Ag+2Bg+2Eg
32-pole (h) 3Au+2Bu+3Eu
64-pole (i) 3Ag+4Bg+3Eg
128-pole (j) 3Au+4Bu+4Eu
256-pole(k) 5Ag+4Bg+4Eg
512-pole (l) 5Au+4Bu+5Eu

First nonvanishing multipole: quadrupole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen