An action on an object which leaves the object at same position after the action carried out. Such type of action is called as Symmetry operations. The simplest example of symmetry operations is water molecule. If we rotate the molecule by 180Â° about an axis which is passing through the central Oxygen atom it will look the same as before. Similarly reflection of molecule through both axis of molecule show same molecule.
There are ï¬ve types of symmetry operations and ï¬ve types of symmetry elements.
1. The identity (E)
This symmetry operation is consists of doing nothing. In other words; any object undergo this symmetry operation and every molecule consists of at least this symmetry operation. For example; bio molecules like DNA and bromo fluoro chloro methane consist of only this symmetry operation. â€˜Eâ€™ notation used to represent identity operation which is coming from a German word â€˜Einheitâ€™ stands for unity.
2. An nfold axis of symmetry (Cn)
 This symmetry operation involves the clockwise rotation of molecule through an angle of 2 Ï€ /n radian where n is an integer. The notation used for nfold of axis is C_{n}.
 For principal axis, the value of n will be highest. The rotation through 360Â°/n angle is equivalent to identity (E).
 For example, one twofold axis rotation of water (H_{2}O) towards oxygen axis leaves molecule at same position, hence has C_{2} axis of symmetry. Similarly ammonia (NH_{3}) has one threefold axis, C_{3} and benzene (C_{6}H_{6}) molecule has one sixfold axis C_{6}and six twofold axis (C_{2}) of symmetry.
Linear diatomic molecules like hydrogen, hydrogen chloride have Câˆž axis as the rotation on any angle remains the molecule the same.
3. Improper rotation (Sn)
Improper clockwise rotation through the angle of 2Ï€/n radians is represented by notation â€˜Snâ€™ and called as nfold axis of symmetry which is a combination of two successive transformations. During improper rotation, the first rotation is through 360Â°/n and the second transformation is a reflection through a plane perpendicular to the axis of the rotation. Improper rotation is also known as alternating axis of symmetry or rotationreflection axis. For example; methane (CH_{4}) molecule has three S_{4} axis of symmetry.
4. A plane of symmetry (Ïƒ)
There are some plane in molecule through which reflection leaves the molecule same. The vertical mirror plane is labelled as Ïƒv and one perpendicular to the axis is called a horizontal mirror plane is labelled as Ïƒh , while the vertical mirror plane which bisects the angle between two C_{2} axes is known as a dihedral mirror plane, Ïƒ_{d}. For example, H_{2}O molecule contains two mirror planes (a YZ Reflection (Ïƒyz) and a XZ Reflection (Ïƒxz)) which are mirror planes contain the principle axis and called as vertical mirror planes (Ïƒ_{v}).
5. Center of symmetry (i)
It is a symmetry operation through which the inversion leaves the molecule unchanged. For example, a sphere or a cube has a centre of inversion. Similarly molecules like benzene, ethane and SF_{6} have a center of symmetry while water and ammonia molecule do not have any center of symmetry.
Overall the symmetry operations can be summarized as given below.
The inversion operation is a symmetry operation which is carried out through a single point, this point is known as inversion center and notated by â€˜ iâ€™. This point is located at the center of the molecule and may or may not coincide with an atom in the molecule.
When we are moving each atom in a molecule along a straight line through the inversion center to a point an equal distance from the inversion center and get same configuration, we say there is an inversion center in the molecule. It can be in such molecules which do not have any atom at center like benzene, ethane.
Geometries like tetrahedral, triangles, pentagons don't contain an inversion center. Hence a cube, a sphere contains a center of inversion but tetrahedron does not contain this symmetry operation. The molecule must be achiral for the presence of inversion center. Some of the common examples of molecules contain center of inversion are as follow.
(a) Benzene molecule: Inversion center located at the center of molecule.
(b) 1,2Dichloroethane: The staggered form of 1,2Dichloroethane contains one inversion center at the center of molecule.
(c) transdiaminedichlorodinitroplatinum complex: trans form of some Coordination compounds like transdiaminedichlorodinitroplatinum complex contains inversion center.
Another example of coordination compound is hexacarbonylchromium complex [Cr(CO)_{6}], where the inversion center located at the position of metal atom in complex.
(d) Ethane molecule: The staggered form of ethane contains inversion center while eclipsed form does not.
(e) Mesotartaric acid: The antiperiplanar conformer of mesotartaric acid has an inversion center.
(f) Dimer of D and LAlanine: The dimer of two configurations of Alanine; Dalanine and Lalanine contains one inversion center.
(g) 18Crown6: An organic compound with the formula [C_{2}H_{4}O]_{6} named as 18crown6 (IUPAC name: 1,4,7,10,13,16hexaoxacyclooctadecane) also contains inversion center located at center of molecule.
(h) Cyclohexane: The chair conformation of cyclohexane contains an inversion center while boat form does not.
A molecule or an object may contains one or more than one symmetry elements, therefore molecules can be grouped together having same symmetry elements and classify according to their symmetry. Such type of groups of symmetry elements are known as point groups because there is at least one point in space which remains unchanged no matter which symmetry operation from the group is applied.
For the labelling of symmetry groups, two systems of notation are given, known as the Schoenflies and HermannMauguin (or International) systems. The Schoenflies notations are used to describe the symmetry of individual molecule. The molecular point groups with their example are listed below.
Point group

Explanation

Example

C_{1}

Contains only identity operation(E) as the C_{1} rotation is a rotation by 360^{o}

Bromochlorofloromethane (CFClBrH) 
C_{i}

Contains the identity (E) and a center of inversion center (i).

Anticonformation of 1,2dichloro1,2dibromoethane.

C_{s}

Contains the identity E and plane of reflection Ïƒ. 
Hypochlorus acid (HOCl), Thionyl chloride (SOCl_{2}). 
C_{n}

Have the identity and an nfold axis of rotation. 
Hydrogen Peroxide (C_{2}) 
C_{nv}

Have the identity, an nfold axis of rotation, and n vertical mirror planes (Ïƒ_{v}).

Water (C_{2v}), Ammonia (C_{3v})

C_{nh}

Have the identity, an nfold axis of rotation, and Ïƒ_{h} (a horizontal reflection plane). 
Boric acid H_{3}BO_{3} (C_{3h}), trans1,2dichloroethane (C_{2h}) 
D_{n}

Have the identity, an nfold axis of rotation with n_{2}fold rotations about the axis which is perpendicular to the principal axis. 
Cyclohexane twist form (D_{2}) 
D_{nh}

Contains the same symmetry elements as D_{n} with the addition of a horizontal mirror plane.

Ethene (D_{2h}), boron trifluoride (D_{3h}), Xenon tetrafluoride (D_{4h}).

D_{nd}

Contains the same symmetry elements as D_{n }with the addition of n dihedral mirror planes. 
Ethane (D_{3d}), Allene(D_{2d}) 
S_{n}

Contains the identity and one Sn axis. 
CClBr=CClBr 
T_{d}

Contains all the symmetry elements of a regular tetrahedron, including the identity, four C3 axis, threeC2 axis, six dihedral mirror planes, and three S4 axis. 
Methane (CH_{4}) 
T Th 
Same as Td but no planes of reflection. Same as for T but contains a center of inversion . 

Oh O

The group of the regular octahedron. Same as Oh but with no planes of reflection. 
Sulphur hexafluoride (SF_{6}) 
Different point groups correspond to certain VSEPR geometry of molecule. Out of them some are as follow.
VSEPR Geometry of molecule

Point group

Linear 
Dâˆžh 
Bent or Vshape

C_{2v}

Trigonal planar

D_{3h }

Trigonal pyramidal

C_{3v }

Trigonal bipyramidal

D_{5h }

Tetrahedral

T_{d}

Sawhorse or seesaw

C_{2v }

Tshape

C_{2v}

Octahedral

Oh

Square pyramidal 
C_{4v}

Square planar

D_{4h}

Pentagonal bipyramidal

D_{5h}

A molecule may contain more than one symmetry operation and show symmetrical nature. Some of the examples of symmetry operation on molecule with their point group are as given below.
(a) Benzene: The point group of benzene molecule is D6h with given symmetry operations.
 Inversion center: i
 The Proper Rotations: seven C_{2} axis and one C_{3} and one C_{6} axis
 The Improper Rotations: S_{6 }and S_{3} axis
 The Reflection Planes: one Ïƒh , three Ïƒ_{v} and three Ïƒ_{d}
(b) Ammonia: The point group of ammonia molecule is C_{3v }with following symmetry operations.
 The Proper Rotations: one C_{3} axis
 The Reflection Planes: three Ïƒ_{v }plane
(c) Cyclohexane: The chair conformation of Cyclohexane has D3d point group with given symmetry operations;
 Inversion center: i
 The Proper Rotations: Three C_{2} axis and one C_{3} axis
 The Improper Rotations: S_{6} axis
 The Reflection Planes: Three Ïƒ_{d} plane
(d) Methane: The point group of methane is T
_{d} (tetrahedral) with C
_{3} as principal axis and other symmetry operations are as follows;
 The Proper Rotations: Three C_{2 }axis and Four C_{3} axis
 The Improper Rotations: Three S_{4} axis
 The Reflection Planes: Five Ïƒ_{d} plane
(e) 12Crown4: This
has S
_{6} point group with C
_{3} and S
_{6} axis with inversion center (i).
(f) Allene: The point group of methane is D
_{2d }with given symmetry operations.
 The Proper Rotations: Three C_{2} axis
 The Improper Rotations: One S_{4} axis
 The Reflection Planes: Two Ïƒ_{d }plane
Group theory deals with symmetry groups which consists of elements and obey certain mathematical laws. Each point group is a set of symmetry operation or symmetry elements which are present in molecule and belongs to this point group. To obtain the complete group of a molecule, we have to include all the symmetry operation including identity â€˜Eâ€™. A character table represents all the symmetry elements correspond to each point group. Hence we can make separate character table for each point group like C_{2v}, C_{3v}, D_{2h}...etc.
For example, in the character table of C_{2v} point group; all the symmetry elements has to written in first row and the symmetry species or Mulliken labels are listed in first column. These symmetry species specify different symmetries within one point group. For C2v, there are four symmetry species or Mulliken labels; A1, A2, B1, B2.
Remember
 The symmetry species for onedimensional representations: A or B
 The symmetry species for twodimensional representations: E
 The symmetry species for threedimensional representations: T
The best example of C2v point group is water which has oxygen as center atom. The px orbital of oxygen atom is perpendicular to the plane of water molecule, hence it is not symmetric with respect to the plane Ïƒv(yz). So this orbital is antisymmetric with respect to the mirror plane and its sign get change when symmetry operations applied. On the other hand, the s orbital is symmetric with respect to mirror plane. The symmetric and antisymmetric nature can be represents by using mathematical sign; +1 and 1; here +1 stands for symmetric and â€“1 stands for antisymmetric which are the characters in character table.
Hence the symmetry operations for the px orbitals are as follow. 
E: Symmetric hence character will be 1

C2: Antisymmetric, hence character will be 1

Ïƒv(xz): Symmetric; character :1

Ïƒv(yz): Antisymmetric, character: 1
Hence the character table for C
_{2v} point group.
C_{2v}

E 
C_{2}

Ïƒ_{v(XZ)}

Ïƒ_{v(YZ)}



A_{1}

1 
1

1 
1 
z

x^{2}, y^{2},z^{2}

A_{2}

1

1

1

1

Rz

xy

B_{1}

1

1

1

1

x, Ry

xz

B_{2}

1

1

1

1

y, Rx

yz

Similarly character can be assigned for other symmetry species. The last two columns of character table make it easier to understand the symmetric nature. For example; x in second last column of B_{1 }symmetry indicates that the xaxis has B_{1 }symmetry in C_{2v }point group and the Rx notation indicates the rotation around the xaxis. Similarly the character table for C_{3v }point group will be
C_{3v}

E

2C_{3}

3_{Ïƒv} 


A_{1}

1 
1 
1 
z

x^{2}+y^{2}, z^{2}

A_{2}

1

1

1

I_{z}


E

2

1

0

(x, y), (I_{xy}, I_{z})

(x^{2}y^{2}, xy), (xz, yz) 
For doubly degenerate, the character for E will be 2 and for triply degenerate it will be 3, because in this case we have two and three orbitals respectively which are symmetric with respect to E. Some of the character tables with their point groups are as follow
a. Character table for Oh point group, for example Sulfur fluorine (SF_{6})
b. Character table Td point group, methane (CH_{4})
c. Character table for D_{3d }point group, for example, staggered ethane
d. Character table for D_{6h
}point group, example Benzene (C_{6}H_{6})
In some molecules like water, ammonia, methane which have more than one symmetry equivalent atom, the combinations of the symmetry equivalent orbitals can transform according to a irreducible representations of the molecules point group which are refer as Symmetry Adapted Linear Combinations. For the formation of an ndimensional representation a set of equivalent functions f1, f2, ..., fn can be used. The representation can be expressed as a sum of irreducible representations with the use of calculation of characters for this representation and by the use of the great orthogonality theorem. The nlinear combinations of f1, ..., fn which transform the irreducible representations is given by the projection operator which denoted as $\hat{p}$ Gi;
Here
 $\hat{p}$ = The operator which projects out of a set of equivalent functions the Gi Irreducible representation of the point group.
 In n/g factor; n = dimension of the irreducible representation
 g = the order of the group
The function fj can be chosen by any one of n which belongs to the equivalent set. For example, in the C_{3v }character table; the 2$\hat{C_3}$_{} represents the class composed by $\hat{C_3^{1}}$ and $\hat{C_3^{1}}$ operations. With C_{3 }class, there are three different $\hat{C_3}$ _{}operations would also perform separately on fj which produce different results. Letâ€™s take an example of the OH stretches along the â€˜yzâ€™ plane as molecular plane in water molecule; the formula can be apply to tabulate the characters of the irreducible representations and list the effect of $\hat{O}$R on one of the functions at the bottom of the table.
C_{2v }

E 
C_{2(Z)}

s_{v(XZ) }

s_{v(YZ) }

A_{1}

1

1 
1 
1 
B_{1}

1

1

+1

1 
B_{2}

1

1

1

+1

OR(OH_{2})

OH_{a}

OH_{b}

OH_{b}

OH_{a}

After applying the projection operator for A
_{1}$\hat{p}$ A_{1} (OHa) = Â¼ (OHa + OHb + OHb + OHa)
=1/2 (OHa + OHb)
According to the orthogonality theorem; it shouldnâ€™t be possible to obtain a B_{1} linear combination, and indeed the projection operator will be zero.
$\hat{p}$ B_{1 }(OHa) = Â¼ (OHa + OHb + OHb + OHa)=0
Application of the B_{2} projection operator gives
$\hat{p}$ B_{2} (OHa) = Â¼ (OHa + OHb + OHb + OHa)
=1/2 (OHa + OHb)
These linear combinations relate to the symmetric (A_{1}) and antisymmetric (B_{2}) stretches of water as given below.
When two equivalent real functions are involved, the correct linear combinations will be equals to the sum and difference functions. In case of degenerate representations like in case of NH stretching vibrations in ammonia, it is more difficult to construct the symmetry adapted linear combinations.
We can create symmetryadapted linear combinations of atomic orbitals in exactly the same way. The point group is D2h with four carbonhydrogen sigma bonds which are symmetryequivalent and can make four carbonhydrogen bonding symmetryAdapted Linear Combinations 's. The character table will be as follow.
Result of symmetry operations on s_{1} 

E 
C_{2(Z)}

C_{2(Y) }

C_{2(X)}

i 
s_{(XY)}

s_{(XZ)}

s_{(YZ)}

s_{1}

s_{1}

s_{3}

s_{4}

s_{2}

s_{3}

s_{1}

s_{2}

s_{4}

There are four nonzero symmetryadapted linear combinations can be possible.