Symmetry in Organic Compounds

Symmetry within organic molecules is a very important concept in terms of understanding their chemical reactivity. In brief, atoms within a molecule which are "identical" will react the same in both chemical reactions and spectroscopy.  There are a number of ways of identifying "identical" atoms within a molecule, these include:

1. Substitution method (simplest but slowest):

The idea is that you replace each H in the molecule in turn with a "dummy" atom (for example a -Cl) to see if you get a different product (i.e. one that will require a different name).  Each new product, indicates a different type of  H. This idea is related to the radical chlorination of alkanes where some of each possible product is usually obtained.

2. Verbal description:

The verbal method requires that you describe the position of each H within the molecule.  If you need to use different words to describe two H atoms, then they represent different types of H.

As examples:

• an -OH is different to a -CH (based on what they are attached to).
• a -CH₃ is different to a -CH₂- (because the number of H at that C are different).
• an sp³ C-H is different to an sp C-H.
• other differences could be position on a chain, across a ring or double bond etc.

3. Symmetry (most difficult but fastest):

The symmetry method is the most sophisticated but the quickest method and requires that you look for mirror planes, rotation axes or inversion centers that interchange atoms.  Atoms that can be interchanged are chemically equivalent to each other.

CAUTION Remember that rotation about σ-bonds produces different conformations only and not different molecules. Thus the three H of a methyl group are always identical. Remember to consider the three dimensional nature of the molecules.

Symmetry Operators

1. Identity operator, E: the identity operator consists of doing nothing to the molecule, therefore all molecules possess this operator, but it achieves nothing.
   

2. Rotation Axis, C_n: if an imaginary line (or axis) can be drawn through a molecule so that rotation by 360°/n gives a molecule indistinguishable from the original, that molecule is said to have a rotation axis, C_n, of order n. Molecules may contain more than one rotation axis, the highest is the principle axis.
   

Examples:
HCN C∞: linear molecule, rotate by any angle. All atoms are unique as the rotation is along the internuclear axis.			HBrClFC C1: rotate by 360°. Note: all C-X bonds are C1 axes. All atoms are unique since a rotation of 360° does not exchange any atoms positions.
CO2 C2: rotate by 180°. Note: the O=C=O axis is a C∞ axis. The two oxygen atoms are identical.			BH3 C3: rotate by 120°. Note: the 3 B-H bonds are also C2 axes. The three H atoms are identical.
Cyclobutane (planar) C4: rotate by 90°. Note: there are four C2 axes perpendicular to the C4 axis, two through opposing faces of the ring, and two through opposing C atoms. All four C atoms are identical, all eight H atoms are identical.			Cyclopentadienyl anion (aromatic) C5: rotate by 72°. Note: there are five C2 axes perpendicular to the C5 axis. All five C atoms are identical, all five H atoms are identical.
Benzene C6: rotate by 60°. Note: there are six C2 axes perpendicular to the C6 axis, three through opposing faces of the ring, and three through opposing C atoms. All six C atoms are identical, all six H atoms are identical.			Cycloheptatrienyl cation C7: rotate by 51.4°. Note: there are seven C2 axes perpendicular to the C7 axis. All seven C atoms are identical, all seven H atoms are identical.

3. Reflection Plane (σ): a molecule has a plane of symmetry if an imaginary double-sided mirror reflects both halves of the molecule into one another so that the new molecule is indistinguishable from the original. In other words the mirror plane divides the molecule into two symmetric halves, each a reflection of the other. Molecules may contain more than one mirror plane. Mirror planes which contain the principle axis are σ_v and those perpendicular to the principle axis are σ_h.  "Diagonal planes", σ_d, are vertical planes that bisect the angles between successive C₂ axes.
   

Examples:
cis-1,2-dichlorocyclopropane mirror plane Note: the atom colors in the animation do NOT represent atom types, they are merely used to clearly indicate the movement of atoms across the mirror plane.			water 2 σv: which contain the principle axis C2 axis Note: only the green σvprovides for symmetry within the molecule (the two H are equivalent). The blue σv is in the plane of the molecule so no atoms are related through this mirror plane.
BH3 σh: perpendicular to principle axis Note: there are σv along each B-H bond perpendicular to the σh plane.			allene (H2C=C=CH2) σd: bisect the angle between two C2 axes.

4. Rotation-Reflection (S_n): a combination of the two previous symmetry elements is a distinct symmetry element called a rotation-reflection or improper rotation axis. This can be described as: S_n  =  C_n  x  σ_h  =  σ_h  x  C_n,  the order of operations is immaterial.
   

Examples:
S4: rotation of 90°, followed by a vertical reflection, or a vertical reflection followed by a 90° rotation.		S2: rotation of 180°, followed by a vertical reflection, or vertical reflection followed by a 180° rotation. NOTE: this corresponds to a center of inversion.

5. Center of Symmetry (or Inversion (i)): a molecule has a center of symmetry if there is a point within the molecule such that reflection of all atoms through that point gives a new molecule which is indistinguishable from the original. The center of symmetry must occur where all rotation axis and mirror planes meet.
   

Examples:
The center of inversion is the "green dot". Draw a straight line from any atom directly to the center of inversion, continue an equal distance in the same direction and you will reach another atom identical to the first. Note: the atoms are colored to clearly indicate the movement for a center of inversion, they do NOT represent atom types.	Another example is 1,4-dibromo-2,5-dichlorocyclohexane, the center of inversion is shown by the blue dot in the center of the molecule.