Advanced Inorganic Chemistry/Symmetry Elements

Advanced Inorganic Chemistry
Symmetry Elements

Advanced Inorganic Chemistry/Symmetry Elements (1.1)

Symmetry elements of the molecule are geometric entities: an imaginary point, axis or plane in space, which symmetry operations: rotation, reflection or inversion, are performed. ^[1],[2] Their recognition leads to the application of symmetry to molecular properties and can also be used to predict or explain many of a molecule’s chemical properties. Symmetry elements and symmetry operations are two fundamental concepts in group theory, which is the mathematical description of symmetry properties that describe the structure, bonding, and spectroscopy of molecules.

Contents

  1. Point of symmetry operations
     1.1. Identity, E
     1.2. Proper Rotation, C_n
     1.3. Reflection, σ
     1.4. Inversion, i
     1.5. Improper Rotation, S_n
  2. Point groups 
  3. Example: symmetry of benzene

1. Point symmetry operations
Point symmetry of a molecule results when there exists at least one point in space that remains indistinguishable from the original molecule after any symmetry operation is applied. In other words, a rotation, reflection or inversion operations are called symmetry operations if, and only if, the newly arranged molecule is indistinguishable from the original arrangement. There are five kinds of point of symmetry elements that a molecule can possess, thus, there are also five kinds of point of symmetry operations. All symmetry elements in a molecule must share at least one point in common and this point occurs at the center of the molecule.

1.1. Identity, E
Identity operation comes from the German ‘Einheit’ meaning unity. This symmetry element means no change. All molecules have this element.

Figure 1. CFClBrI. A chiral molecule possesses no symmetry but identity operation, E.

1.2. Proper Rotation, C_n
Proper rotation operates with respect to an axis called a symmetry axis (also known as n-fold rotational axis). An axis around which a rotation by 360°/n (or 2π/n) results in an identical molecule before and after the rotation. The axis with the highest n is called the principal axis.

Figure 2. Octahedral structure of SF₆. Proper rotations of SF₆ are 5C₂, 4C₃ and 3C₄ and we call C₄ the principal axis of SF₆.

In general, a molecule contains nC_n operations such that {C_n¹, C_n², C_n³,…, C_n^n-1, C_nⁿ} where C_nⁿ = E. For example, if there exists C₅ axis then there exists 5C₅ (2C₅¹, 2C₅², C₅⁵) operations:

C₅¹ = C₅⁴ since the operation results indistinguishable molecule in clockwise and counter-clockwise, respectively.
C₅² = C₅³ since the operation results indistinguishable molecule in clockwise and counter-clockwise, respectively.
C₅⁵ = E

1.3. Reflection, σ
Reflection operates with respect to a plane called a plane of symmetry (also known as a mirror plane). There exist three types of mirror planes:

σ_h – a horizontal mirror plane of a molecule is perpendicular to the primary axis of a molecule.
σ_v – a vertical mirror plane of a molecule includes the primary axis of a molecule and passes through the bonds (atoms).
σ_d – a dihedral (also known as a diagonal mirror plane) of a molecule includes the primary axis of a molecule while bisecting the angle between two C₂ axes that are perpendicular to it. Therefore, ơ_d does not pass through the bonds (atoms).

NOTE: σ_d is a special case of a σ_v.

Figure 3. Bent structure of H₂O. The water molecule has two σ_v along C₂ axis and no σ_h.

1.4. Inversion, i
Inversion operates with respect to a point called a center of symmetry (also known as an inversion center). It gives the same result to rotating a molecule around C₂ axis then reflecting it with respect to a mirror plane that is perpendicular to C₂. For example, SF₆ in figure 2 has an inversion point at the center S.

1.5. Improper Rotation, S_n
Improper rotation operates with respect to an axis called rotation-reflection axis. In other words, it is a combination operation of a rotation about an axis by 360°/n (or 2π/n) followed by reflection in a plane perpendicular to the rotation axis.

Figure 4. Staggered ferrocene (left) has an S₁₀ operation while eclipsed ferrocene (right) has an S₅ operation.

2. Point groups
The complete collection of symmetry elements of a molecule forms the basis of a mathematical group, and the collection of symmetry operations that are interrelated to each other via certain kind of rules is known as a point group:

Closure – if any two symmetry operations are in the same group then their product, resulting in another operation, will also be in the same group:

                                    If A∈G and B∈G, then (A∩B)∈G

Associativity – the law of associativity applies to all symmetry operations:

                                          (AB)C=A(BC)

Identity – there exists an operation that commutes with other operations (identity, E) and leaves them unchanged:

                                      If A∈G and E∈G, then AE=EA=A

Inverses – for every symmetry operation in the group, there exists an inverse operation that their product results identity:

                       If A∈G, then there exists A^-1∈G such that AA^-1=A^-1A=E

For more about mathematical representation of symmetry properties, see also:

Molecular Point Group (1.2)
Matrices (1.3)
Representations (1.4)

in Advanced Inorganic Chemistry.

3. Example: symmetry of benzene
Benzene is one of the molecules that possess various symmetry elements and symmetry operations.

4. References
[1] Pfenning, Brian W. (2015). Principles of Inorganic Chemistry. Hoboken: John Wiley & Sons, Inc.. pp. 195.
[2] https://www-e.openu.ac.il/symmetry/symmetry-tutorial.html