Symmetry Elements and Operations

• 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

Activity 1

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.

1. BrF₅
2. ferrocene (eclipsed)
3. S₈ (puckered)
4. 1,5-dibromonaphthalene
5. Ni(cyclobutadiene)₂ (staggered)
6. CH₄
7. CHCl₃
8. CH₂Cl₂
9. acetylene
10. cis- and trans-[Cu(gly₂)]
11. cis- and trans-1,2-dichloroethylene (DCE)

Return to Chemistry, UWI-Mona, Home Page