Symmetry Elements and Symmetry Operation (Works hand in hand)

Thirumohoor
The Art of Symmetry in Architecture

A structure of balanced proportion is usually termed symmetry. Symmetries of objects are detailed by 

1. Symmetry elements

2. Symmetry operation

Symmetry Elements are the geometrical entities such as a line, a plane, an axis to which one can perform one or more symmetry operations say reflection, rotation, inversion yielding an indistinguishable configuration.

Symmetry Element: Identity Element (analogous to 1 in multiplication operation)

Symmetry Operation: Identity Operation

Representation: E

Every molecule possesses an identity element. Identity Operation is not really an operation since it corresponds to rotation through an angle of 360° (equivalent to doing nothing 0°). However, used to define the term inverse.

Symmetry Element: Centre of Symmetry or Inversion Centre or Centro symmetric

Symmetry Operation: Inversion

Representation: i

If every atom in a molecule is identical to the diametrically opposite atom, then the molecule is said to have an inversion center. The centrosymmetric molecule undergoes inversion operation i.e., the linear movement of atoms through the inversion center. (Illustration: Cyclobutane)

Symmetry Element: Axis of Symmetry

Symmetry Operation: Rotation

Representation: Cn

With respect to the axis of symmetry, rotation through 360°/n (=θ) allows us an orientation that can't be distinguished from the original. Cn generates n operations (C_n¹, C_n², C_n³, …, C_nⁿ (= E)) 

Description: θ is the angle of rotation

                     n is the order of rotation (n= 2 fold/ 3 fold/ 4 fold axis of symmetry...)

Also, a molecule can have more than one order of rotation. If this statement becomes true for a molecule, the axis of symmetry with the highest order is regarded as the Principal axis (Illustration: Cyclobutane) and all others are subsidiary axis (Illustration: Cyclobutane). Both principal and subsidiary axes are reciprocally perpendicular.

Symbol of Proper Rotation Axis (Cn)	Order of Proper Rotation Axis (n)	θ= 360°/n
C2	2	180°
C3	3	120°
C4	4	90°
C5	5	72°
C6	6	60°

Math Results:

C_nⁿ = E

C_nⁿ⁺¹ = C_n

C_nⁿ⁺² = C_n²...

Symmetry Element: Plane of Symmetry

Symmetry Operation: Reflection

Representation: 𝛔

If the mirror plane reflects the identical atoms, the molecule holds a plane of symmetries (Horizontal plane of symmetry, Vertical plane of symmetry, Diagonal or dihedral plane of symmetry)

The mirror plane perpendicular to the principal axis is referred to as the horizontal plane of symmetry (𝜎h) (Illustration: Cyclobutane)

The mirror plane consisting principal axis is referred to as the vertical plane of symmetry (𝜎v) (Illustration: Cyclobutane)

The mirror plane with the principal axis and bisects the angle between a pair of C2 axis of symmetry is referred to as the dihedral or diagonal plane of symmetry (𝜎d) (Illustration: Cyclobutane)

Priority Order: 𝜎h > 𝜎v > 𝜎d

Symmetry Element: Improper Axis of Symmetry

Symmetry Operation: Rotation followed by Reflection

Representation: Sn= Cn.𝛔h

This operation is a mingle of rotation through 360°/n (=θ) and reflection through the mirror plane perpendicular to the principal axis. (Illustration: Cyclobutane). Like Cn, Sn also generates n operations. 

Illustration: 

n=1 (odd),

S₁¹= C₁. 𝜎_h= E. 𝜎_h= 𝜎_h     

S₁²= C₁². 𝜎_h²= C₁ C₁. E= E E. E =E

n=2 (even),

S₂¹= C₂. 𝜎_h= i  (Illustration: Trans-1,2-difluoroethylene)    

S₂²= C₂². 𝜎_h²= E. E= E

n=3 (odd),

S₃¹= C₃. 𝜎_h (Illustration: Eclipsed Ethane, Boron trifluoride)

S₃²= C₃². 𝜎_h²= C₃². E= C₃²

S₃³= C₃³. 𝜎_h³= C₃³. 𝜎_h² 𝜎_h= E. E 𝜎_h= 𝜎_h

S₃⁴= C₃⁴. 𝜎_h⁴= C₃³ C₃. 𝜎_h² 𝜎_h²= E C₃. E E= C₃

S₃⁵= C₃⁵. 𝜎_h⁵= C₃³ C₃². 𝜎_h² 𝜎_h² 𝜎_h = E C₃². E E 𝜎_h = C₃². 𝜎_h

S₃⁶= C₃⁶. 𝜎_h⁶= C₃³ C₃³. 𝜎_h² 𝜎_h² 𝜎_h²= E E. E E E= E

n=4 (even),

S₄¹= C₄. 𝜎_h (Illustration: Methane)

S₄²= C₄². 𝜎_h²= C₄². E= C₄²

S₄³= C₄³. 𝜎_h³= C₄³. 𝜎_h² 𝜎_h= C₄³. E 𝜎_h= C₄³. 𝜎_h

S₄⁴= C₄⁴. 𝜎_h⁴= C₄⁴. 𝜎_h² 𝜎_h²= E. E E= E

n=5 (Illustration: Eclipsed Ferrocene) 

n=6 (Illustration: Staggered Ethane)

n=10 (Illustration: Staggered Ferrocene) 

If n is even, Sn generates n operations. If n is odd, Sn generates 2n operations.