2.MOLECULAR POINT GROUPS

Successive operations 

as we have already studied about some operation for the symmetry in molecule there are mainly five operation which are as follow 

1. axes of symmetry
2. Plane of symmetry
3. Improper rotational axis
4. Center of symmetry
5. Identity element

there are few more operation which are the combination and the repetition of the above operation which brings in its identity such operations are known as successive operations.

For Example 

1. In Ammonia NH3 possesses a C3 axis tells us that we can rotate the molecule through 1200 and end up with a molecular configuration that is indistinguishable from the first. However, it takes three such operations to give a configuration of the NH3 molecule that exactly coincides with the first. The three separate 1200 rotations are identified by using the notation in Figure . We cannot actually distinguish between the three H atoms, but for clarity they are labelled H(1), H(2) and H(3) in the figure. Since the third rotation,  C₃³, returns the NH3 molecule to its initial configuration.

C₃³  = E for ammonia

and C_nⁿ  = E  in general 

2. In BCl₃ the operation C₃ followed by the operation σ_h shows the equivalent.

Point groups

The symbolic representation for a molecule which shows the number and the nature of symmetric elements present in a molecule is known as the Point group. 

Some point groups are as follow

1. 1.    C_s            = E, having only plane of symmetry
   2.       C_i             = E, Inversion Center
   3.       C_n            = E, principal n-fold axis of symmetry
   4.       C_nv          = E, principal n-fold axis of symmetry and n vertical plane of symmetry
   5.       C_nh          = E, principal n-fold axis of symmetry and one horizontal plane of symmetry.
   6.       D_nh          = E, principal n-fold axis of symmetry and nC₂ and one horizontal plane of symmetry.
   7.       D_nd          = E, principal n-fold axis of symmetry and nC₂ and one dihedral plane of symmetry.
   8.       T_d            = Tetrahedral
   9.       O_h           = Octahedral
   10.   I_h             = Icosahedral

C₁ point Group

Molecule that appear no symmetry at all. except E is known as C_1.

C∞v point group

A molecule which is linear and having ∞-fold axis of rotation are known as C∞v. It must also possesses infinite no. of vertical plane of symmetry and not any horizontal plane of symmetry. some asymmetric diatomic molecule or ions HF, CO, [CN]^- and linear polyatomic are the example of such type of point group.

D∞h Point group

symmetric diatomic (H₂, [O₂]^{- )}and linear polyatomic molecule (CO₂, [N₃]^-)  that contain center of symmetry and possesses horizontal plane of symmetry in addition to a C∞ and infinite vertical plane of symmetry.

Note in the above two cases the sign ∞ indicates there is no limit of angle of rotation it will always show the symmetry about any value of n for Cn .

Explanation for the determination of point group

Determine the point groups of the following molecule

1. Trans- N₂F₂
2. PF₅

Solution

1.  the structure of Trans- N₂F₂ 

Is the molecule linear?                                             No

Does it have Td, Oh or Ih symmetry?                        No

Is there a Cn axis?                                                    Yes; a C2 axis perpendicular to

                                                                                 the plane of the paper and passing 

                                                                                 through the midpoint of the N-N bond.

Are there two C2 axes perpendicular

to the principal axis?                                                No

Is there a σh plane                                                    Yes 

So the point group is C2h

Solution

2. The structure of PF5

Is the molecule linear?                                             No

Does it have Td, Oh or Ih symmetry?                        No

Is there a Cn axis?                                         Yes; a C3 axis containing the P and two axial 
                                                                      F atom
Are there two C2 axes perpendicular
to the principal axis?                                    Yes; each lie along a P- Feq
Is there a σh plane                                        Yes; it contains the P and three Feq atoms 


the point group is D3h

Scheme for assigning point groups of molecules and molecular ions. Apart from the cases of n= 1 or ∞, n most commonly has values of 2, 3, 4, 5 or 6.

it can explain by the following picture

Classification of molecules into point group

On the basis of the definition of point group thee molecules are classified into three following categories 

1. Molecule of Low Symmetry (MLS)
2. Molecule of High Symmetry (MHS)
3. Molecule of Special Symmetry (MSS)

Molecule of Low Symmetry (MLS)

The molecule which contain only few or no symmetry elements are known as molecules of low symmetry

It may have one of the following characteristics

1. Molecules just containing E e.g. CHFClBr, SOClBr.
2. Molecules containing just E, σ e.g. OHD, 1,2,4 - trisubstituted benzene and aniline 
   
3. Molecule containing just E,i e.g. C3H4Cl2Br2 
   

Molecule of High Symmetry (MHS)

This class of molecule would invariably contain a Cn axis. The presence of of other type of symmetry elements along with Cn will determine the nature of point group. Molecules falling under this category are plenty and they will be considered under Cn, Dn and Sn type of point groups. Such type of molecules show in three categories of MHS.

Cn type point group 

In this class, the molecule must contain only one type of rotational axis (Cn). where n > 2. Depending on the nature of some more additional symmetry elements, the molecules can be further grouped into the following types.

Cn molecular point groups

The molecule containing only Cn axis and no other elements belong to this category of point groups. Since the presence of Cn axis implies the presence of n distinct elements including E, the order of this point group is n. This is a pure rotational group since this contains only rotational axes.

H₂O₂ is an example of such type of point group belongs to C2 point group. The C2 axis passes through the mid point of O-O bond as shown in fig.

C_nv molecular point groups

Which shows Cn + nσv 
Example NH3, H2O type of molecule

Cnh this type of point group also included in such category, which we have already studied earlier.

Dn type of point groups

The molecule containing a combination of Cn +nC2  belong to Dn category. Such categories of molecule is furthermore classified as follow

Dn molecular point group

Cn+nC2 only

D_nh molecular point groups

Which refers

                                C_n + nC_n + sigma_h

Example

Benzene

D_nd molecular point groups

Which refers

C_n + nC₂+ sigma_d

Example

Staggered ethane is an example of D_nd.

1,3,5,7- tetramethylcyclooctatetraene

Molecules of Special symmetry (MSS)

Under this category, consider two types of unique molecular structures:

Linear molecules

Perfect polyhedral molecules (platonic geometries)

Linear molecules

They belong to infinite groups since they all contain a C_infinite. The presence and absence of certain other elements, in addition to the C_infinite axis would classify the molecule into th following categories.

1. C_infinite + ∞sigma₂ ⇨C_(infinite)v               all these molecules are linear but asymmetrical in nature. They may be diatomic or polyatomic type but with unsymmetrical distribution of atoms or groups. Simplest examples of molecules in this category are CO, CN, HCl, NO, SCN, HCN or even the following type that we have discussed earliear.

2. C_infinite + ∞C₂ + σ_h    ⇨  D_(infinite)v

Already discussed.

Perfect polyhedral molecules

This is a special type of molecules containing multiple higher order axes (mC_n with n≥3; m is even )

1- Molecules containing multiple C₃ axes (T_d, T_h, T)-

Td point groups: