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):

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):

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^o/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:
	Axis off/on			Animation on/off Axis off/on Spin off/on
HCN C∞: linear molecule, rotate by any angle. All atoms are unique as the rotation is along the internuclear axis.			HBrClFC C1: rotate by 360o. Note: all C-X bonds are C1 axes. All atoms are unique since a rotation of 360o does not exchange any atoms positions.
	Animation on/off Axis off/on Spin off/on			Animation on/off Axis off/on Spin off/on
CO2 C2: rotate by 180o. Note: the O=C=O axis is a C∞ axis. The two oxygen atoms are identical.			BH3 C3: rotate by 120o. Note: the 3 B-H bonds are also C2 axes. The three H atoms are identical.
	Animation on/off Axis off/on Spin off/on			Animation on/off Axis off/on Spin off/on
Cyclobutane (planar) C4: rotate by 90o. Note: there are eight C2 axes perpendicular to the C4 axis, four through opposing faces of the ring, and four through opposing C atoms. All four C atoms are identical, all eight H atoms are identical.			Cyclopentadienyl anion (aromatic) C5: rotate by 72o. Note: there are five C2 axes perpendicular to the C5 axis. All five C atoms are identical, all five H atoms are identical.
	Animation on/off Axis off/on Spin off/on			Animation on/off Axis off/on Spin off/on
Benzene C6: rotate by 60o. 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 51.4o. 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:
	Animation on/off Plane off/on			σv plane off/on σv plane off/on C2 axis off/on
cis-1,2-dichlorocyclopropane mirror plane			water 2 σv: which contain the principle axis C1 axis Note: only the green σv provides 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.
	σh plane off/on C3 axis off/on			σd planes off/on C2 axes off/on
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. This can be described as:
   S_n  =  C_n  x  σ_h  =  σ_h  x  C_n,  the order of operations is immaterial.
   

Examples:
Animation on/off S4 axis off/on		Animation on/off S2 axis off/on
S4: rotation of 90o, followed by a vertical reflection, or a vertical reflection followed by a 90o rotation.		S2: rotation of 180o, followed by a vertical reflection, or vertical reflection followed by a 180o 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:
Animation on/off
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.	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.