Character table for point group D7h

D7h E 2C7 2(C7)2 2(C7)3 7C'2 h 2S7 2(S7)5 2(S7)3 7v
linear functions,
rotations
quadratic
functions
cubic
functions
A'1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A'2 +1 +1 +1 +1 -1 +1 +1 +1 +1 -1 Rz - -
E'1 +2 +2cos(2/7) +2cos(4/7) +2cos(6/7) 0 +2 +2cos(2/7) +2cos(4/7) +2cos(6/7) 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E'2 +2 +2cos(4/7) +2cos(6/7) +2cos(2/7) 0 +2 +2cos(4/7) +2cos(6/7) +2cos(2/7) 0 - (x2-y2, xy) -
E'3 +2 +2cos(6/7) +2cos(2/7) +2cos(4/7) 0 +2 +2cos(6/7) +2cos(2/7) +2cos(4/7) 0 - - [y(3x2-y2), x(x2-3y2)]
A''1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -
A''2 +1 +1 +1 +1 -1 -1 -1 -1 -1 +1 z - z3, z(x2+y2)
E''1 +2 +2cos(2/7) +2cos(4/7) +2cos(6/7) 0 -2 -2cos(2/7) -2cos(4/7) -2cos(6/7) 0 (Rx, Ry) (xz, yz) -
E''2 +2 +2cos(4/7) +2cos(6/7) +2cos(2/7) 0 -2 -2cos(4/7) -2cos(6/7) -2cos(2/7) 0 - - [xyz, z(x2-y2)]
E''3 +2 +2cos(6/7) +2cos(2/7) +2cos(4/7) 0 -2 -2cos(6/7) -2cos(2/7) -2cos(4/7) 0 - - -


Additional information

Number of symmetry elements h = 28
Number of irreducible representations n = 10
Abelian group no
Number of subgroups8
Number of distinct subgroups7
Subgroups
(Number of different orientations)
Cs (2) , C2 , C7 , D7 , C2v , C7v , C7h
Optical Isomerism (Chirality) no


Reduction formula for point group D7h

Type of representation

general 3N vib

E 2C7 2(C7)2 2(C7)3 7C'2 h 2S7 2(S7)5 2(S7)3 7v




Examples

Tropylium Cation



Multipoles

dipole (p) E'1+A''2
quadrupole (d) A'1+E'2+E''1
octopole (f) E'1+E'3+A''2+E''2
hexadecapole (g) A'1+E'2+E'3+E''1+E''3
32-pole (h) E'1+E'2+E'3+A''2+E''2+E''3
64-pole (i) A'1+E'1+E'2+E'3+E''1+E''2+E''3
128-pole (j) A'1+A'2+E'1+E'2+E'3+A''2+E''1+E''2+E''3
256-pole(k) A'1+2E'1+E'2+E'3+A''1+A''2+E''1+E''2+E''3
512-pole (l) A'1+A'2+E'1+2E'2+E'3+A''2+2E''1+E''2+E''3

First nonvanishing multipole: quadrupole

Literature



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