Character table for point group D5d

(x axis coincident with C'2 axis)
D5d E 2C5 2(C5)2 5C'2 i 2(S10)3 2S10 5d
linear functions,
rotations
quadratic
functions
cubic
functions
A1g +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2g +1 +1 +1 -1 +1 +1 +1 -1 Rz - -
E1g +2 +2cos(2/5) +2cos(4/5) 0 +2 +2cos(2/5) +2cos(4/5) 0 (Rx, Ry) (xz, yz) -
E2g +2 +2cos(4/5) +2cos(2/5) 0 +2 +2cos(4/5) +2cos(2/5) 0 - (x2-y2, xy) -
A1u +1 +1 +1 +1 -1 -1 -1 -1 - - -
A2u +1 +1 +1 -1 -1 -1 -1 +1 z - z3, z(x2+y2)
E1u +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)]
E2u +2 +2cos(4/5) +2cos(2/5) 0 -2 -2cos(4/5) -2cos(2/5) 0 - - [xyz, z(x2-y2)] [y(3x2-y2), x(x2-3y2)]

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
Subgroups Cs , Ci , C2 , C5 , D5 , C5v , C2h , S10
Optical Isomerism (Chirality) no


Reduction formula for point group D5d

Type of representation

general 3N vib

E 2C5 2(C5)2 5C'2 i 2(S10)3 2S10 5d




Multipoles

dipole (p) A2u+E1u
quadrupole (d) A1g+E1g+E2g
octopole (f) A2u+E1u+2E2u
hexadecapole (g) A1g+2E1g+2E2g
32-pole (h) A1u+2A2u+2E1u+2E2u
64-pole (i) 2A1g+A2g+3E1g+2E2g
128-pole (j) A1u+2A2u+3E1u+3E2u
256-pole(k) 2A1g+A2g+3E1g+4E2g
512-pole (l) A1u+2A2u+4E1u+4E2u

First nonvanishing multipole: quadrupole

Literature



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