Symmetry and the point groups. Symmetry Elements and Symmetry Operations. Identity Proper axis of rotation Mirror planes Center of symmetry Improper axis of rotation. Symmetry Elements and Symmetry Operations. Identity => E. Symmetry Elements and Symmetry Operations.
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sh => mirror plane perpendicular to a
principal axis of rotation
sv => mirror plane containing principal
axis of rotation
sd => mirror plane bisects dihedral angle made
by the principal axis of rotation and two
adjacent C2 axes perpendicular to principal
1. determine the highest axis of rotation
2. check for other non-coincident axis of rotation
3. check for mirror planes
Point Groups with improper axes
S2n (n ≥ 2)
In contrast to groups C, D, and S, cubic symmetry groups are characterized by the presence of several rotational axes of high order (≥ 3).
Cases of regular polyhedra:
Symmetry elements:E, 4C3, 3C2, 3S4, 6sd
Symmetry operations: E, 8C3, 3C2, 6S4, 6sd
If all planes of symmetry and i are missing, the point group is T (pure rotational group, very rare);
If all dihedral planes are removed but 3 sh remain, the point group is Th ( [Fe(py)6]2+ )
Symmetry elements:E, i, 4S6, 4C3, 3S4, 3C4, 6C2, 3 C2, 3sh, 6sd
Symmetry operations: E, i, 8S6, 8C3, 6S4, 6C4, 6C2, 3 C2, 3sh, 6sd
Pure rotational analogue is the point group O (no mirror planes and no Sn; very rare)
Th group (symmetry elements: E, i, 4S6, 4C3, 3C2, 3sh) can also be considered as a result of reducing Ohgroup symmetry (E, i, 4S6, 4C3, 3S4, 3C4, 6C2, 3 C2, 3sh, 6sd )
by eliminating C4, S4 and some C2 axes andsd planes
Symmetry elements: E, i, 6S10, 6C5, 10S6, 10C3, 15C2, 15s
Pure rotation analogue is the point group I (no mirror planes and thus no Sn, very rare)
A molecule may be chiral only if it does not posses an axis of improper rotation Sn.
All molecules with center of inversion are achiral optically inactive.
S1=s any molecule with a mirror plane is achiral.