Molecular Geometry & Bonding
what the shape decides
Ladder

Representation — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. Tdorder 24E1 found8C38 found3C23 found6S46 found6σd6 foundA111111A2111-1-1E2-1200T130-11-1T2x, y, z30-1-115 representations for 5 classes · dimensions² sum to 24rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated24 operations

    Character tables and reduction

    A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be.

    rung 1 · symmetry
  2. Tdfrom ↓ to →A1A2ET1T2A1————xyzA2———xyz—E———xyzxyzT1—xyzxyzxyzxyzT2xyz—xyzxyzxyz12 allowed, 13 forbidden of 25a dash is exactly zero, not merely small — the integrand cancels in pairsone theorem, applied to every pairletters give the polarisation

    Selection rules are one theorem

    An integral over all space vanishes unless the integrand is totally symmetric. Every selection rule in spectroscopy is that sentence with a different integrand — and the rule of mutual exclusion falls out rather than being remembered.

    rung 2 · symmetry
  3. FFFPFFD3hprincipal axis C34 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates6 atoms

    Site symmetry, and what it constrains

    A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.

    rung 3 · symmetry

All ladders