Molecular Symmetry and Group Theory

Workshop 2 — Point groups

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  • Exercise 2

    The full set of symmetry operations of a molecule constitutes a mathematical group. To form a group, the operations must satisfy four rules:

    1. The product of any 2 members (and the square of any one) must also be a member of the group
    2. There must be an identity member E
    3. For any 3 operations (A, B, C) performed successively, the associative law is obeyed i.e. (AB)C = A(BC)
    4. Every member A has an inverse A-1 also in the group, such that AA-1 = E

      Use the multiplication table obtained in Workshop 1 (shown below) for C2H4O (C2v) to demonstrate the validity of these rules.
    [For iii) use A = C2, B = σ(xz), C = σ(yz).]

    C2v E C2 σ(xz) σ(yz)
    E E C2 σ(xz) σ(yz)
    C2 C2 E σ(yz) σ(xz)
    σ(xz) σ(xz) σ(yz) E C2
    σ(yz) σ(yz) σ(xz) C2 E


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