HOW TO FIND OUT POINT GROUP Dr. Cyriac Mathew Symmetry elements can combine in a definite number of ways. For example consider BF3 molecule. It has the C3 axis as the principal axis. Also, there are 3 C2 axes perpendicular to the C3 axis, 3 σv planes, and one σh plane. All these symmetry elements can combine. Take NH3. It also has C3 axis as principal axis. There are 3 σv planes, but no C2 perpendicular to C3 or a σh plane. For the N2F2 molecule there exist one C2 axis and a σh plane but no σv planes. Thus in general we can say a Cn can combine with either nC2 or no C2 perpendicular to it; it can combine with either one σh or no σh; or it can combine with n vertical planes or no vertical planes. These are relevant only in the case of systems where we can identify a principal axis. Such systems are called axial systems. In the case of tetrahedral, octahedral, cubic, icosahedral and dodecahedral objects we cannot identify the principal axis. Solids of these structures are called platonic solids. Crystals and molecules of these shapes are highly symmetric and can be called multi higher order axial systems. They have several higher order axes than C2 axis. The symmetry elements combine in definite ways in these systems also. Point Groups Point groups are possible combinations of symmetry elements. Since symmetry elements can combine only in a definite pattern, there will be only a finite number of point groups possible. For crystals only 32 points groups exist. Crystals cannot have axes of symmetry order 5 or higher than 6. On the other hand molecules can have proper axes of symmetry of order 5, 7 and ∞ also. Hence for molecules some additional point groups are possible which are not possible for crystals. Redundant symmetry elements: symmetry elements which combine to form a point group are known as essential symmetry elements. There exist some symmetry elements which are present as a consequence of the essential symmetry elements. Consider BF3 molecule. The essential symmetry elements needed, for the point group under which this molecule comes, are E, C3, 3σv, σh. The presence of C3 and σh give rise to S3 (ie. σhC3). The following table provides the possible point groups for molecules and crystals. The notation given is the Schoenflies notation which is applicable to molecules. In the case of crystals the Hermann-Mauguin notation is used. Table 1: Molecular point groups Point group (Schoenflies notation) Cn Cs Ci Cnh Cnv Dn Dnh Essential symmetry elements Only a proper axis Cn Only a plane of symmetry Only centre of symmetry A proper axis Cn and σh A proper axis Cn and n σv A principal axis Cn and nC2 perpendicular to that Sn T Th Td O Oh 3C4, 4C3 , 6C2 and 3 σh Sn, nσv (i for even values of n) A proper axis Cn, nC2 perpendicular to that and σh A proper axis Cn, nC2 perpendicular to that, and nσd Only an Sn 4C3 and 3C2 4C3 , 3C2 and σh or i 4C3 , 3C2 and 6σd 3C4, 4C3 , and 6C2 Dnd redundant symmetry elements Sn - S2n - Values of n for molecules Values of n for crystals 1,2,3,4,5,6 1,2,3,4,6 2,3,4,5,6,7,∞ 2,3,4,5,6,7, ∞ 2,3,4,6 2,3,4,6 2,3,4,5,6 2,3,4,6 2,3,4,5,6,7, ∞ 2,3,4,6 2,3,4,5 2,3 2,4,6 2,4,6 3S4 3S4, 4S6, 6 σv and i Cyclic point groups (Cn,Cnv, Cnh and Sn): A molecule which possesses only a Cn axis come under the Cn point group. Here C stands for cyclic and n stands for order of the axis. Cn represents five groups, namely, C1, C2, C3, C4, C5 and C6. C5 is a non-crystallographic point group. C1 is the point group with no element of symmetry. H O O O Br F Cl bromochlorofluoro methane C1 point group H H H Hydrogen peroxide Phosphoric acid C2 point group C3 point group C6 point group C5(CH3)5. pentamethyl cyclopentadienyl C5 point group S2 C6(CH3)6. hexamethyl cyclohexadienyl P O H O O H N Quinoline Cs point group Br Cl Ci point group i Cl Br Assigning Point Group First of all we look for an axis of symmetry and find whether it is the principal axis or not. There can be molecules without an axis of symmetry at all and corresponding point groups are C i, Cs, S4 and C1only. If axes of symmetry are there and the principal axis cannot be identified, the point group can be Td, Oh or Ih. If there is a Cn axis and nC2 axes perpendicular to it the point group will be dihedral, Dn, Dnh, Dnd. Otherwise, the point groups will be Cn, Cnh, or Cnv. Thus the following steps are involved in assigning the point group. Identify, 1. The presence of an axis of symmetry 2. This axis as principal axis or not 3. The existence of subsidiary axes 4. The existence of σh 5. The presence of n σv’s 6. Whether belongs to Ci, Cs, S4 or C1, if principal axis is absent. This procedure can be simplified by asking certain questions to ourselves and finding ‘yes’ or ‘no’ answers. Is there an axis of symmetry? (Find the highest order axis) Yes No Is it the principal axis? Is there a plane of symmetry? No Yes Is there an i? No No Cs Look for Td and Oh point groups Yes Are there nC2’s perpendicular Cn? Yes Is there an Sn? Yes No Ci Is there a σh? No Yes Sn Are there nσv? No Yes Cn Cnh Are there nσd? No Cnv Yes No Yes No C1 Is there a σh? Dn Yes Dnd Dnh

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