Lecture 14 APPLICATIONS OF GROUP THEORY 1) Symmetry of atomic orbitals

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Presentation transcript:

Lecture 14 APPLICATIONS OF GROUP THEORY 1) Symmetry of atomic orbitals When bonds are formed, atomic orbitals combine according to their symmetry. Symmetry properties and degeneracy of orbitals can be learned from corresponding character tables by their inspection. Hold in mind the following transformational properties: Atomic orbital Transforms as s x2+y2+z2 px x py y pz z dz2 z2, 2z2-x2-y2 dx2-y2 x2-y2 dxy xy dxz xz dyz yz

2) Examples of atomic orbitals symmetry analysis C2v A1 z x2, y2, z2 A2 Rz xy B1 x, Ry xz B2 y, Rx yz Atomic orbital Mulliken labels C2v D3h D4h Td Oh s px py pz dz2 dx2-y2 dxy dxz dyz D3h A1’ x2+y2, z2 A2’ Rz E’ (x,y) (x2-y2, xy) A1” A2” z E” (Rx,Ry) (xz, yz) D4h A1g x2+y2, z2 B1g x2-y2 B2g xy Eg (Rx,Ry) (xz, yz) A2u z Eu (x, y) Td A1 x2+y2+z2 A2 E (2z2-x2-y2, x2-y2) T1 (Rx,Ry,Rz) T2 (x,y,z) (xz, yz, xy) Oh A1g x2+y2+z2 Eg (2z2-x2-y2, x2-y2) T1g (Rx,Ry,Rz) T2g (xz, yz, xy) T1u (x,y,z) …

3) Symmetry adapter linear combinations of atomic orbitals (SALC’s) Hybrid orbitals can be considered as basis functions for a reducible representation Gr within a molecule point group. Let us choose vectors originating from the central atom to represent the hybrid orbitals suitable for s-bonding as a basis function for Gr. When constructing a reducible representation Gr, we have to consider the effect of each of the group symmetry operations on these vectors. The character of a particular symmetry operation is equal to the number of vectors that are unshifted by the operation.

4) Symmetry adapted linear combinations of AO’s for s-bonding D4h E 2C4 C2 2C2’ 2C2” i 2S4 sh 2sv 2sd Gr Td E 8C3 3C2 6S4 6sd Gr D3h E 2C3 3C2 sh 2S3 3sv Gr-axial Gr-equ

5) SALC’s of atomic orbitals suitable for p-bonding Let us choose a set of vectors originating from the peripheral atoms and representing directions of the hybrid orbitals suitable for p-bonding with the central atom as a basis function for Gr. All vectors xi are directed toward z axis and all vectors yi are parallel to xy plane. Only vectors of unshifted atoms contribute to the character of particular symmetry operations. Td E 8C3 3C2 6S4 6sd Gr

6) Polarity A species of high symmetry (several rotational axes) cannot be polar. The polarity is a feature of molecules belonging to the following symmetry point groups only: C1, Cs, Cn, Cnv.