Question

Show that a molecule with an inversion center implies the presence of an \(S_{2}\) element.

Short answer

To demonstrate that a molecule with an inversion center implies the presence of an \(S_2\) element, follow these steps: 1. Locate the inversion center of the molecule. 2. Choose any two symmetrically inverted atoms, A and B. 3. Rotate the molecule by \(180^\circ\) about an axis perpendicular to the line joining atoms A and B, resulting in the alignment of symmetrically inverted partners. 4. Reflect the positions of all atoms in a plane perpendicular to the C2 rotation axis and containing the inversion center, swapping the positions of symmetrically inverted atom pairs. 5. Compare the final positions and original positions. If they are the same, this demonstrates the presence of an \(S_2\) element.

Step-by-step solution

Locate the inversion center

Determine the inversion center of the molecule by finding the point through which all atoms are symmetrically inverted. This point may or may not be an atom.

Choose any two symmetrically inverted atoms

Select any two atoms (A and B) such that A is at distance \(r\) on one side of the inversion center and B is at the same distance \(r\) on the opposite side of the inversion center.

Perform the C2 rotation

Rotate the molecule by \(180^\circ\) about an axis perpendicular to the line joining atoms A and B and passing through the inversion center. Notice that atom A aligns with atom B, and atom B aligns with atom A, meaning all other atoms are also aligned with their symmetrically inverted partners.

Perform the σ reflection

Reflect the positions of all atoms in a plane perpendicular to the C2 rotation axis and containing the inversion center. This reflection swaps the positions of atoms A and B, as well as every other pair of symmetrically inverted atoms.

Compare the final positions and original positions

After performing the sequence of transformation operations, compare the molecules' final positions to their original positions. If the final positions are the same as the original positions, we have demonstrated the presence of an \(S_2\) element. In conclusion, by following these steps, we can show that a molecule with an inversion center implies the presence of an \(S_2\) element.

Key concepts

Inversion Center

An inversion center is a crucial concept in understanding molecular symmetry. It is a point within a molecule through which all parts of the molecule are symmetrically opposite. If you draw a straight line from any atom to the inversion center, the same line will extend to find a symmetrically identical atom on the opposite side.

This imaginary center need not coincide with an actual atom in the molecule. Its existence ensures that the molecule can be turned inside out, similar to flipping an image but in a three-dimensional manner. Having an inversion center is an indication of a specific symmetry operation that a molecule can perform, known as inversion.

Molecular Symmetry

Molecular symmetry refers to the spatial arrangement of atoms in a molecule that remains unchanged, even after certain operations or transformations. These symmetrical arrangements are not only aesthetically pleasing but important in determining the properties and behaviors of molecules.

Symmetrical molecules often have properties like reduced polarity or certain optical behaviors. Recognizing symmetry within molecules helps scientists predict how they will react with other substances. Symmetry elements, including inversion centers, rotation axes, and reflection planes, define how these transformations are observed in molecules.

Rotation Operations

Rotation operations in molecular geometry involve rotating a molecule around an imaginary axis. These axes, known as rotation axes, pass through the molecule and create a scenario where the molecule appears unchanged after a complete rotation.

The most common rotational axes are expressed as C",

Reflection Operations

Reflection operations involve flipping a molecule over a mirror plane to produce an identical structure. In this context, a mirror plane is an imaginary surface bisecting the molecule such that each half is the mirror image of the other.

Therefore, reflection involves swapping atom positions across this plane. In the presence of inversion centers and rotational axes, reflection operations contribute to the identification of more complex symmetry elements like the S_2 operation, which combines rotation and reflection.