## Symmetry Elements

The symmetry of a molecule can be described by 5 types of symmetry elements.
• Symmetry axis: an axis around which a rotation by 360/n results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system.
• Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σd). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz).
• Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF4) where the inversion center is at the Xe atom, and benzene (C6H6) where the inversion center is at the center of the ring.
• Rotation-reflection axis: an axis around which a rotation by 360/n, followed by a reflection in a plane perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn. Examples are present in tetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis.
• Identity, abbreviated to E, from the German 'Einheit' meaning unity. This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, its consideration is necessary for the group-theoretical machinery to work properly. It is so called because it is analogous to multiplying by one (unity).

## Objectives

This exercise consists of two activities.
Activity 1: To use software to draw and manipulate structures in order to identify the generators and determine their Point Groups.
Activity 2: To Use Group Theory and Spectroscopy to identify the isomers of bisglycinatocopper(II) hydrate (Cu(gly)2.H2O)

# Activity 1

Using Argus Lab, draw each molecule.

Then, by studying the molecule in various orientations, identify the generators and use the flowchart to predict the point group. Consult the character tables and list the remaining symmetry operations in each group. Use diagrams to show the location of all symmetry elements.

The molecules to be studied include:
1. BrF5
2. ferrocene (eclipsed)
3. S8 (puckered)
4. 1,5-dibromonaphthalene
6. CH4
7. CHCl3
8. CH2Cl2
9. acetylene
10. cis- and trans-[Cu(gly2)]
11. cis- and trans-1,2-dichloroethylene (DCE)

# Activity 2a

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 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 2b

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.

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.

 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(ethylenediamine)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-

## C2v Point Group

Abelian, 4 irreducible representations
Subgroups of C2v point group: Cs, C2

Character table for C2v point group

E C2 (z) σv(xz) σv(yz) linear,
rotations