Symmetry in Chemistry - Group Theory

Symmetry inChemistry - Group Theory

Group Theory is one of the most prevailing mathematical tools used in Quantum Chemistry and Spectroscopy. It permits the user to forecast, interpret, reduce, and often abridge complex theory and data.
At its heart is the fact that the Set of Operations related with the Symmetry Elements of a molecule comprise a mathematical set known as a Group. This permits the application of the mathematical theorems allied with such groups to the Symmetry Operations.

All Symmetry Operations related with isolated molecules can be distinguished as Rotations:

(a) Proper Rotations: C_n^k ; k = 1,......, n 
When k = n, C_n^k = E, the Identity Operation 
n designate a rotation of 360/n where n = 1,....

(b) Improper Rotations: S_n^k , k = 1,....., n 
When k = 1, n = 1 S_n^k = s , Reflection Operation 
When k = 1, n = 2 S_n^k = i , Inversion Operation

In general practice we distinguish Five types of operation:

(i) E, Identity Operation

(ii) C_n^k , Proper Rotation about an axis

(iii) s, Reflection through a plane

(iv) i, Inversion through a center

(v) S_n^k, Rotation about an axis followed by indication through a plane perpendicular to that axis.

Each of these Symmetry Operations is linked with a Symmetry Element which is a point, a line, or a plane concerning which the operation is performed such that the molecule's orientation and position prior to and after the operation are interchangeable.

The Symmetry Elements linked with a molecule are:-

Symmetry Elements

(i) A Proper Axis of Rotation: C_n where n = 1,.... 
This implies n-fold rotational symmetry about the axis.

(ii) A Plane of Reflection: s 
This implies bilateral symmetry concerning the plane. 
These planes are further confidential as:

s_h - Horizontal Plane which is perpendicular to the Principal Axis of Rotation (i.e. Axis with highest value of n). If no principal axis exists s_h is distinct as the molecular plane.

s_v or s_d - Vertical Plane which holds the Principal Axis of Rotation and is perpendicular to a s_h plane, if it exists. When mutually s_v and s_d planes are present, the s_v planes hold the superior number of atoms, the s_d planes enclose bond angle bisectors. If only one type of vertical plane is present, s_v or s_d may be used depending on the entirety symmetry of the molecule.

(iii) A Center of Inversion - i 
This is a central point throughout which all C_n and s elements have to pass. If no such ordinary point exists there is refusal center of symmetry.

(iv) Improper Axis: S_n 
This is invented of two parts: C_n and s_h both of which may or may not be factual symmetry elements of the molecule. If equally the C_n and s_h are there then S_n must also exist. The following relations are obliging in this stare: 
(a) If n is even, S_nⁿ = E 
(b) If n is odd, S_nⁿ = s and S_n²ⁿ = E 
(c) If m is even, S_n^m = C_n^m when m < n 
S_n^m = C_n^m-n when m > n 
(d) If S_n with even n exists then C_n/2 exists. 
(e) If S_n with odd n exists then both C_n and s perpendicular to C_n exist.