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Molecular Symmetry

Any object is called as symmetrical if it has mirror symmetry, or ‘left-right’ symmetry i.e. it would look the same in a mirror. For example, a sphere is more symmetrical compare to a cube as it looks the same after rotation through any angle about the diameter.    However during the rotation of a cube, it looks similar only with certain angles like 90°, 180°, or 270° about an axis passing from the centers of any of its opposite faces, or by 120° or 240° about an axis passing from any of the opposite corners.
In the same way, symmetry is very important in chemistry as some molecules are clearly ‘more symmetrical’ than others due to their different configurations. For example; some of the molecules like NH3, H2O, C6H6, CBrClF remain same after certain symmetry operations like inversion, rotations and reflections. Each symmetry operation corresponds to symmetry element which can be a plane, point or line with respect to performed symmetry operation.
All known molecules can classify in groups possess the same set of symmetry elements, and same set of symmetry elements. Such type of classi?cation is helpful to assign the molecular properties without calculation and in the determination of polarity and degeneracy of molecular states. It provides the systematic treatment of symmetry in chemical systems in a mathematical framework which is called as group theory.

Group theory is also helpful is some other investigations
  1. Prediction of polarity and chirality of molecule.
  2. In examination of bonding and visualizing molecular orbitals of molecules.
  3. In prediction of polarization a molecule.
  4. In investigation of vibrational motions of the molecule.

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Symmetry Operations

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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 180° 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.

Symmetry Operations

There are five types of symmetry operations and five 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. ‘E’ notation used to represent identity operation which is coming from a German word ‘Einheit’ stands for unity.

2. An n-fold 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 n-fold of axis is Cn.
  • For principal axis, the value of n will be highest. The rotation through 360°/n angle is equivalent to identity (E).
  • For example, one twofold axis rotation of water (H2O) towards oxygen axis leaves molecule at same position, hence has C2 axis of symmetry. Similarly ammonia (NH3) has one threefold axis, C3 and benzene (C6H6) molecule has one sixfold axis C6and six twofold axis (C2) of symmetry.

n-Fold Axis of Symmetry

Linear diatomic molecules like hydrogen, hydrogen chloride have C∞ axis as the rotation on any angle remains the molecule the same.

3. Improper rotation (Sn)

Improper clockwise rotation through the angle of 2π/n radians is represented by notation ‘Sn’ and called as n-fold axis of symmetry which is a combination of two successive transformations. During improper rotation, the first rotation is through 360°/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 rotation-reflection axis. For example; methane (CH4) molecule has three S4 axis of symmetry.
Improper Rotation

4. A plane of symmetry (σ)

There are some plane in molecule through which reflection leaves the molecule same. The vertical mirror plane is labelled as σv and one perpendicular to the axis is called a horizontal mirror plane is labelled as σh , while the vertical mirror plane which bisects the angle between two C2 axes is known as a dihedral mirror plane, σd. For example, H2O molecule contains two mirror planes (a YZ Reflection (σyz) and a XZ Reflection (σxz)) which are mirror planes contain the principle axis and called as vertical mirror planes (σv).
Plane of Symmetry

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 SF6 have a center of symmetry while water and ammonia molecule do not have any center of symmetry.
Center of Symmetry

Overall the symmetry operations can be summarized as given below.

Symmetry Operations

Inversion Center

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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 ‘ i’. 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) 1,2-Dichloroethane: The staggered form of 1,2-Dichloroethane contains one inversion center at the center of molecule.

(c) trans-diaminedichlorodinitroplatinum complex: trans- form of some Coordination compounds like trans-diaminedichlorodinitroplatinum 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.
Hexacarbonylchromium Complex
(d) Ethane molecule: The staggered form of ethane contains inversion center while eclipsed form does not.

Ethane Molecule

(e) Meso-tartaric acid: The anti-periplanar conformer of meso-tartaric acid has an inversion center.

Meso Tartaric Acid
(f) Dimer of D and L-Alanine: The dimer of two configurations of Alanine; D-alanine and L-alanine contains one inversion center.
Dimer of D and L-Alanine

(g) 18-Crown-6: An organic compound with the formula [C2H4O]6 named as 18-crown-6 (IUPAC name: 1,4,7,10,13,16-hexaoxacyclooctadecane) 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.

Molecular Symmetry Examples

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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 Hermann-Mauguin (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
C1 Contains only identity operation(E) as the C1 rotation is a rotation by 360o Bromochlorofloromethane (CFClBrH)
Contains the identity (E) and a center of inversion center (i).
Anti-conformation of 1,2-dichloro-1,2-dibromoethane.
Cs Contains the identity E and plane of reflection σ. Hypochlorus acid (HOCl), Thionyl chloride (SOCl2).
Cn Have the identity and an n-fold axis of rotation. Hydrogen Peroxide (C2)
Have the identity, an n-fold axis of rotation, and n vertical mirror planes (σv).
Water (C2v), Ammonia (C3v)
Cnh Have the identity, an n-fold axis of rotation, and σh (a horizontal reflection plane). Boric acid H3BO3 (C3h), trans-1,2-dichloroethane (C2h)
Dn Have the identity, an n-fold axis of rotation with n2-fold rotations about the axis which is perpendicular to the principal axis. Cyclohexane twist form (D2)
Contains the same symmetry elements as Dn with the addition of a horizontal mirror plane.
Ethene (D2h), boron trifluoride (D3h), Xenon tetrafluoride (D4h).
Dnd Contains the same symmetry elements as Dn with the addition of n dihedral mirror planes. Ethane (D3d), Allene(D2d)
Sn Contains the identity and one Sn axis. CClBr=CClBr
Td Contains all the symmetry elements of a regular tetrahedron, including the identity, four C3 axis, three-C2 axis, six dihedral mirror planes, and three S4 axis. Methane (CH4)
Same as Td but no planes of reflection.
Same as for T but contains a center of inversion .
The group of the regular octahedron.
Same as Oh but with no planes of reflection.
Sulphur hexafluoride (SF6)

Different point groups correspond to certain VSEPR geometry of molecule. Out of them some are as follow.

VSEPR Geometry of molecule
Point group
Linear D∞h
Bent or V-shape
Trigonal planar
Trigonal pyramidal
Trigonal bipyramidal
Sawhorse or see-saw
Square pyramidal C4v
Square planar
Pentagonal bipyramidal

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 C2 axis and one C3 and one C6 axis
  • The Improper Rotations: S6 and S3 axis
  • The Reflection Planes: one σh , three σv and three σd
Benzene Molecule

(b) Ammonia: The point group of ammonia molecule is C3v with following symmetry operations.
  • The Proper Rotations: one C3 axis
  • The Reflection Planes: three σv plane

(c) Cyclohexane:
The chair conformation of Cyclohexane has D3d point group with given symmetry operations;
  • Inversion center: i
  • The Proper Rotations: Three C2 axis and one C3 axis
  • The Improper Rotations: S6 axis
  • The Reflection Planes: Three σd plane
(d) Methane: The point group of methane is Td (tetrahedral) with C3 as principal axis and other symmetry operations are as follows;
  • The Proper Rotations: Three C2 axis and Four C3 axis
  • The Improper Rotations: Three S4 axis
  • The Reflection Planes: Five σd plane

(e) 12-Crown-4: This has S6 point group with C3 and S6 axis with inversion center (i).
(f) Allene: The point group of methane is D2d with given symmetry operations.
  • The Proper Rotations: Three C2 axis
  • The Improper Rotations: One S4 axis
  • The Reflection Planes: Two σd plane

Molecular Symmetry and Group Theory

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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 ‘E’. 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 C2v, C3v, D2h...etc.

For example, in the character table of C2v 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.

  • The symmetry species for one-dimensional representations: A or B
  • The symmetry species for two-dimensional representations: E
  • The symmetry species for three-dimensional 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 σv(yz). So this orbital is anti-symmetric 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 anti-symmetric nature can be represents by using mathematical sign; +1 and -1; here +1 stands for symmetric and –1 stands for anti-symmetric which are the characters in character table.
Hence the symmetry operations for the px orbitals are as follow.
  1. E: Symmetric hence character will be 1
  2. C2: Anti-symmetric, hence character will be 1
  3. σv(xz): Symmetric; character :1
  4. σv(yz): Anti-symmetric, character: -1
Hence the character table for C2v point group.

E C2
A1 1 1
1 1 z
x2, y2,z2
x, Ry
y, Rx

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 B1 symmetry indicates that the x-axis has B1 symmetry in C2v point group and the Rx notation indicates the rotation around the x-axis. Similarly the character table for C3v point group will be

A1 1 1 1 z
x2+y2, z2
(x, y), (Ixy, Iz)
(x2-y2, 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 (SF6)

Character Table for OH Point Group
b. Character table Td point group, methane (CH4)

Character Table Td Point Group
c. Character table for D3d point group, for example, staggered ethane

Character Table for D3d Point Group
d. Character table for D6h point group, example Benzene (C6H6)

Character Table for D6h Point Group

Symmetry Adapted Linear Combinations

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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 n-dimensional 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 n-linear combinations of f1, ..., fn which transform the irreducible representations is given by the projection operator which denoted as $\hat{p}$ Gi;

Symmetry Adapted
  • $\hat{p}$ = 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 C3v character table; the 2$\hat{C_3}$ represents the class composed by $\hat{C_3^{1}}$ and $\hat{C_3^{1}}$ operations. With C3 class, there are three different $\hat{C_3}$ operations would also perform separately on fj which produce different results. Let’s take an example of the O-H stretches along the ‘yz’ 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.

E C2(Z)
A1 1
1 1 1

After applying the projection operator for A1

$\hat{p}$ A1 (O-Ha) = ¼ (O-Ha + O-Hb + O-Hb + O-Ha)
=1/2 (O-Ha + O-Hb)
According to the orthogonality theorem; it shouldn’t be possible to obtain a B1 linear combination, and indeed the projection operator will be zero.

$\hat{p}$ B1 (O-Ha) = ¼ (O-Ha + O-Hb + O-Hb + O-Ha)=0

Application of the B2 projection operator gives

$\hat{p}$ B2 (O-Ha) = ¼ (O-Ha + O-Hb + O-Hb + O-Ha)
=1/2 (O-Ha + O-Hb)

These linear combinations relate to the symmetric (A1) and anti-symmetric (B2) stretches of water as given below.

Symmetric (A1) and Anti-symmetric (B2)

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 N-H stretching vibrations in ammonia, it is more difficult to construct the symmetry adapted linear combinations. 

N-H Stretching Vibrations
Symmetry Adapted Linear Combinations

We can create symmetry-adapted linear combinations of atomic orbitals in exactly the same way. The point group is D2h with four carbon-hydrogen sigma bonds which are symmetry-equivalent and can make four carbon-hydrogen bonding symmetry-Adapted Linear Combinations 's. The character table will be as follow.

Result of symmetry operations on s1
E C2(Z) C2(Y)
i s(XY)
s1 s1
s3 s4
s2 s3 s1
s2 s4

There are four non-zero symmetry-adapted linear combinations can be possible. 
Four Non-Zero Symmetry
More topics in Molecular Symmetry
NH3 Molecular Geometry CH4 Molecular Geometry
H2O Molecular Geometry CO2 Molecular Geometry
SO3 Molecular Geometry XeF2 Molecular Geometry
XeF4 Molecular Geometry
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