What is Symmetry?

Symmetry in Chemistry

Symmetry usually conveys two most important meanings. The first is an indefinite sense of harmonious or aesthetically pleasant proportionality and balance; such that it imitates beauty or perfection. The second meaning is a specific and definite idea of balance or "patterned self-similarity" that can be established or proved according to the policy of a ceremonial system: by geometry and physics.

Point Group Symmetry

Point Group Symmetry

Point group symmetry is a significant property of molecules extensively used in several branches of chemistry: spectroscopy, quantum chemistry and crystallography.
An entity point group is characterizing through a set of symmetry operations:

• E - the individuality operation
• C_n - rotation by 2π/n angle *
• S_n - improper rotation (rotation by 2π/n angle and indication in the plane perpendicular to the axis)
• σ_h - horizontal reflection plane (perpendicular to the principal axis) **
• σ_v - vertical reflection plane (encloses the principal axis)
• σ_d - diagonal reflection plane (encloses the principal axis and bisect the angle among two C₂ axes perpendicular to the principal axis)

* - n is an integer
** - principal axis is a C_n axis with the largest n.

Molecule belongs to a symmetry point group if it is untouched under all the symmetry operations of this group.

Symmetric Elements and Chirality

The existence of a single chiral center point to chirality’s of a molecule. a new way to know chirality is to detect the existence of some symmetry elements.

A molecule will not be chiral if it haves.:-

• A plane of symmetry (s)
• A center of symmetry 
• An n fold alternating axis of symmetry (S_n or C_n) where n is an even number.
• These three are known as elements of symmetry.

Plane of symmetry

Plane of symmetry

A plane that separates an object into two equal halves is known as plane of symmetry. Its also known as mirror plane at the same time as it cuts a molecule into two parts, where one is the mirror image of the other. Molecules having such a plane are for all time inactive because of inside compensation. For e.g., meso tartaric acid has a plane of symmetry.

Mesotartaric acid

There are two symmetric carbon atoms.  They are consequently achiral. Therefore meso tartaric acid is optically inactive.

Center of symmetry

Center of symmetry

The center of symmetry is an imaginary point in the molecule. If a line is drained from an atom or a group of the molecule to this imaginary point and then comprehensive to an equal distance ahead of the point, it meets the mirror image of the atom or group. For e.g., trans-1, 4-dimethyl-diketopiperazine has a axis of symmetry and consequently optically inactive.

If a line is strained from methyl group on carbon 1 to the center of symmetry and comprehensive away from this point by an equal distance it meets the methyl group at carbon 4, consequently it is optically inactive.

Alternating axis of symmetry

Alternating axis of symmetry

When a structure is spin throughout an angle of 2p/n about an imaginary axis and then reflects across a plane perpendicular to the axis, an identical composition results.

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.