Lecture 7: Symmetry Elements and Operations

Read section 3.1, 3.2 from your textbook and read the information on this page, then take your next online quiz. Quiz 4 is due by Tuesday, January 26 at 10 PM.

Symmetry operations act upon symmetry elements. Symmetry operations allow us to see the relationships among equivalent atoms in a molecule. Atoms within a molecule are equivalent when they are interconverted by passing through a mirror plane, rotating around an axis passing through the molecule, or passing through an inversion center.

We can use the set of symmetry operations of complex molecules to determine their symmtry point group. This will aid us in formulating molecular orbital diagrams to understand their bonding.

Mirror Plane

  • If there is a plane that passes through a molecule such that one side of the plane is a mirror image of the other, that plane is a mirror plane ().
  • The operation of exchanging the atoms in the molecule through the mirror plane is a reflection.
  • There can be multiple mirror planes in the same molecule

Let's look at a simple example you're probably familiar with from your organic chemistry class.

Here is a picture of meta-xylene, a disubstituted benzene ring.


The carbon centers are shown here. This is, of course, and aromatic molecule and the C-C bond distances in the benzene ring are equivalent. We're only going to consider the carbon atoms for the moment.
Note that the molecule is planar so the plane of the molecule is a mirror plane. We label this main plane of the molecule .

However, since there are no carbon atoms above or below this plane, no carbon atoms are equivalent through this horizontal plane.


There is another mirror plane that bisects the molecule. This horizontal plane is labeled to distinguish it from the vertical plane, .

The carbon atoms symmetrically disposed across the mirror plane are equivalent. This molecule has 5 kinds of carbon atoms.

Proper Rotation Axis

  • If a molecule has a line that passes through it and, if rotation around that line by some fraction of 360 degrees gives a structure identical to the original structure, the line is a proper rotation axis, C.
  • The act of rotating around a proper rotation axis is the symmetry operation of rotation.
  • The order of a rotation axis is the largest number n such that rotation by 360/n degrees gives the original structure. The axis is designated Cn.
  • There can be multiple proper rotation axes in a molecule.
  • The highest order rotation axis Cn is the axis in a molecule with the largest number n.
  • There can be multiple rotations around an axis. For example, if there is a C3 axis, rotation by 360/3 or 120 deg and rotation by (2)(360)/3 or 240 deg are symmetry operations.



Imagine a rod passing through C1 and C5. This is a C2 rotation axis.

If the molecule rotates by 360/2 or 180 degrees, it is indistinguishable from the original molecule. The two methyl groups, for example, are made equivalent through this rotation.

Inversion Center

  • If you can pass each part of a molecule through a point in the center and the resulting structure is identical to the original structure, the molecule has an inversion center, i.
  • The operation of passing the parts of a molecule through such a center is the operation of inversion.



Trans-difluorocyclobutane is shown here.



Imagine a point at the center of the C4 ring. This is the inversion center. All atoms can pass straight through this point to equivalent positions on the other side of the molecule. The fluorine atoms are equivalent because of this inversion.

Improper Rotation Axis

An improper rotation requires both an axis and a plane perpendicular to it.
  • If a molecule has a line that passes through it and, if rotation around that line by some fraction of 360 degrees followed by reflection through a horizontal plane gives a structure identical to the original structure, the line is an improper rotation axis, S.
  • The act of rotating around an improper rotation axis is the symmetry operation of improper rotation.
  • The order of a rotation axis is the largest number n such that rotation by 360/n degrees followed by reflection gives the original structure. The axis is designated Sn.
  • There can be multiple improper rotation axes in a molecule.
  • The highest order improper rotation axis Sn is the axis in a molecule with the largest number n.
  • There can be multiple improper rotations around an axis.



An inversion center is the same as an improper rotation axis of order 2 (S2). Imagine an axis passing through the molecule, perpendicular to the C4 plane. This is an S2 axis. If you rotate the molecule 180 degress then pass all atoms though a plane perpendicular to the rotation axis (in this case the C4 plane) you get an identical molecule.



There is an S6 plane in cyclohexane. If you rotate by 360/6 degrees then pass all molecules through a perpendicular plane, you get an identical structure.

Identity

The identity operator is designated as E. This operation leaves the molecule unchanged so every molecule has that operator.
  • Any 2 reflections returnes the molecule to its original state:
      x = E
  • Any 2 inversions returnes the molecule to its original state:
      i x i = E
  • n proper rotations around a Cn axis returnes the molecule to its original state, for example:
      C3 x C3 x C3 = E

Hydrogen

Let's reexamine the symmetry of H2, a very high symmetry molecule.



Rotation axes

  • There is a rotation axis that is coincedent with the bond axis. Rotation by any number of degrees about this axis give an identical molecule. In other words, there are an infinite number of degrees n for this rotation axis C360/n. We designate this as C
  • There are in infinite number of C2 axes perpendicular to the principle rotation axis at the mid-point of the H-H bond. Rotation by 360/2 degrees on any of these gives the same molecule.
  • The proper rotation axes are also improper rotation axes in this very high symmetry molecule.

Mirror planes

  • There are an infinite number of mirror planes that each include the C axis and one of the C2 axes.
  • There is a mirror plane perpendicular to the bond axis, a , that includes all of the C2 axes.

Inversion center

    There is an inversion center at the mid-point of the H-H bond.

Identity

    This molecule, like all others, has an identity operator.