Molecular Symmetry and Group Theory

Workshop 5 — Chemical bonding

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  • Exercise 4

    In the lecture we used group theory to generate a qualitative MO energy diagram for the H2O molecule. To determine appropriate symmetries for the hydrogen orbitals we generated a reducible representation (Γ) using a 1s orbital on each of the two hydrogen atoms as a basis. Similarly in Exercise 3 we used a 1s orbital on each of the four H atoms in CH4 as a basis for generating Γ.

    Suppose we attempt to generate a representation of the symmetry operations of the C2v point group using a hydrogen 1s orbital on only one of the hydrogen atoms.

    C2v E C2 σv(xz) σv′(xz)    
    A1 1 1 1 1 z x2,y2,z2
    A2 1 1 -1 -1 Rz xy
    B1 1 -1 1 -1 x,Ry xz
    B2 1 -1 -1 1 y,Rx yz

    C2v E C2 σ σ′
    H 1s

    Click for larger image
    Is this an irreducible representation of the C2v point group? (yes/no)
    Can the representation be reduced? (yes/no)

    A single H 1s orbital in H2O is therefore not an appropriate basis for a representation of the C2v point group. As the two H 1s orbitals are equivalent by symmetry in the water molecule, they must be treated together and we must generate a reducible representation by considering both orbitals.

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