Point group Symmetry elements Simple description, chiral if applicable Illustrative species
C1 E no symmetry, chiral CFClBrH, lysergic acid
Cs E σh mirror plane, no other symmetry thionyl chloride, hypochlorous acid
Ci E i Inversion center anti-1,2-dichloro-1,2-dibromoethane
C∞v E 2C∞ σv linear hydrogen chloride, dicarbon monoxide
D∞h E 2C∞ ∞σi i 2Sσ ∞C2 linear with inversion center dihydrogen, azide anion, carbon dioxide
C2 E C2 "open book geometry," chiral hydrogen peroxide
C3 E C3 propeller, chiral triphenylphosphine
C2h E C2 i σh planar with inversion center trans-1,2-dichloroethylene
C3h E C3 C32 σh S3 S35 propeller Boric acid
C2v E C2 σv(xz) σv'(yz) angular (H2O) or see-saw (SF4) water, sulfur tetrafluoride, sulfuryl fluoride
C3v E 2C3 3σv trigonal pyramidal ammonia, phosphorus oxychloride
C4v E 2C4 C2 2σv 2σd square pyramidal xenon oxytetrafluoride
D2 E C2(x) C2(y) C2(z) twist, chiral cyclohexane twist conformation
D3 E C3(z) 3C2 triple helix, chiral tris(1,2-diaminoethane)cobalt(III) cation
D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz) planar with inversion center ethylene, dinitrogen tetroxide, diborane
D3h E 2C3 3C2 σh 2S3 3σv trigonal planar or trigonal bipyramidal boron trifluoride, phosphorus pentachloride
D4h E 2C4 C2 2C2' 2C2 i 2S4 σh 2σv 2σd square planar xenon tetrafluoride
D5h E 2C5 2C52 5C2 σh 2S5 2S53 5σv pentagonal ruthenocene, eclipsed ferrocene, C70 fullerene
D6h E 2C6 2C3 C2 3C2' 3C2 i 3S3 2S63 σh 3σd 3σv hexagonal benzene, bis(benzene)chromium
D2d E 2S4 C2 2C2' 2σd 90° twist allene, tetrasulfur tetranitride
D3d E C3 3C2 i 2S6 3σd 60° twist ethane (staggered rotamer), cyclohexane chair conformation
D4d E 2S8 2C4 2S83 C2 4C2' 4σd 45° twist dimanganese decacarbonyl (staggered rotamer)
D5d E 2C5 2C52 5C2 i 3S103 2S10 5σd 36° twist ferrocene (staggered rotamer)
Td E 8C3 3C2 6S4 6σd tetrahedral methane, phosphorus pentoxide, adamantane
Oh E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd octahedral or cubic cubane, sulfur hexafluoride
Ih E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ icosahedral C60, B12H122-



Character table for C2v point group

E C2 (z) σv(xz) σv(yz) linear,
rotations		 quadratic
A1 1 1 1 1 z x2, y2, z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz

Character table for C2h point group

E C2 (z) i σh
linear,
rotations
 quadratic
Ag 1 1 1 1 Rz x2, y2, z2, xy
Bg 1 -1 1 -1 Rx, Ry xz, yz
Au 1 1 -1 -1 z
Bu 1 -1 -1 1 x, y


At the end of this exercise you should be able to:
  1. Use ACDLabs/Arguslab to draw chemical structures.
  2. Find the generators and determine the point group of any molecule.
  3. Determine which irreducible representation of a Point Group labels the symmetry of a particular molecular vibration.
  4. Predict the number of bands expected in the IR or Raman spectra of simple molecules.
  5. Use SGT to differentiate between cis and trans isomers.

Activity 2a

To identify the observed and calculated differences between the spectra of the isomers of bis-(glycinato)copper(II).hydrate (Cu(gly)2).H2O, based on Group Theory.
You are provided with the IR and Raman spectra for the cis- and trans- isomers of Cu(gly)2.H2O. Identify the major differences in the spectra between the cis- and the trans- isomers. Using Group Theory, and showing all reasoning, rationalise these differences.

Activity 2b

To identify the observed and calculated differences between the spectra of the isomers of 1,2-dichloroethylene (DCE) based on Group Theory.
You are provided with the IR and Raman spectra for the DCE. Identify the major differences in the spectra between the cis- and the trans- isomers. Using Group Theory, and showing all reasoning, rationalise these differences.

Activity 2c

To identify the observed differences between the H and D (deuterated) spectra of various chloromethanes and calculate the expected frequency changes based on the change of mass.

Acknowledgements.
Much of the information in these notes has been sourced from Wikipedia under the Creative Commons License.
http://www.webqc.org/symmetry.php