Character table for point group D5h

(x axis coincident with C'2 axis)
D5h E 2C5 2(C5)2 5C'2 h 2S5 2(S5)3 5v
linear functions,
rotations
quadratic
functions
cubic
functions
A'1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A'2 +1 +1 +1 -1 +1 +1 +1 -1 Rz - -
E'1 +2 +2cos(2/5) +2cos(4/5) 0 +2 +2cos(2/5) +2cos(4/5) 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E'2 +2 +2cos(4/5) +2cos(2/5) 0 +2 +2cos(4/5) +2cos(2/5) 0 - (x2-y2, xy) [y(3x2-y2), x(x2-3y2)]
A''1 +1 +1 +1 +1 -1 -1 -1 -1 - - -
A''2 +1 +1 +1 -1 -1 -1 -1 +1 z - z3, z(x2+y2)
E''1 +2 +2cos(2/5) +2cos(4/5) 0 -2 -2cos(2/5) -2cos(4/5) 0 (Rx, Ry) (xz, yz) -
E''2 +2 +2cos(4/5) +2cos(2/5) 0 -2 -2cos(4/5) -2cos(2/5) 0 - - [xyz, z(x2-y2)]

Information for point groups with fivefold rotational axis


Additional information

Number of symmetry elements h = 20
Number of irreducible representations n = 8
Abelian group no
Number of subgroups8
Number of distinct subgroups7
Subgroups
(Number of different orientations)
Cs (2) , C2 , C5 , D5 , C2v , C5v , C5h
Optical Isomerism (Chirality) no


Reduction formula for point group D5h

Type of representation

general 3N vib

E 2C5 2(C5)2 5C'2 h 2S5 2(S5)3 5v




Examples

Cyclopentadenyl Anion



Multipoles

dipole (p) E'1+A''2
quadrupole (d) A'1+E'2+E''1
octopole (f) E'1+E'2+A''2+E''2
hexadecapole (g) A'1+E'1+E'2+E''1+E''2
32-pole (h) A'1+A'2+E'1+E'2+A''2+E''1+E''2
64-pole (i) A'1+2E'1+E'2+A''1+A''2+E''1+E''2
128-pole (j) A'1+A'2+E'1+2E'2+A''2+2E''1+E''2
256-pole(k) A'1+2E'1+2E'2+A''1+A''2+E''1+2E''2
512-pole (l) A'1+A'2+2E'1+2E'2+A''2+2E''1+2E''2

First nonvanishing multipole: quadrupole

Literature



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