## 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 quadraticfunctions cubicfunctions 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 - - -

 Cs (2) , C2 , C7 , D7 , C2v , C7v , C7h Number of symmetry elements h = 28 Number of irreducible representations n = 10 Abelian group no Number of subgroups 8 Number of distinct subgroups 7 Subgroups (Number of different orientations) Optical Isomerism (Chirality) no Polar 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

## Multipoles

dipole (p) E'1+A''2 A'1+E'2+E''1 E'1+E'3+A''2+E''2 A'1+E'2+E'3+E''1+E''3 E'1+E'2+E'3+A''2+E''2+E''3 A'1+E'1+E'2+E'3+E''1+E''2+E''3 A'1+A'2+E'1+E'2+E'3+A''2+E''1+E''2+E''3 A'1+2E'1+E'2+E'3+A''1+A''2+E''1+E''2+E''3 A'1+A'2+E'1+2E'2+E'3+A''2+2E''1+E''2+E''3