This preview shows page 1. Sign up to view the full content.
Unformatted text preview: e perpendicular C2 axis (speciﬁcally the C2 containing outer atoms): If it is 1 then the subscript on the A or B is 1 If it is  1 then the subscript on the A or B is 2 Note: A diﬀerent set of rules apply to symmetry labels E and T. These won’t be covered. 3. If there are no perpendicular C2 axes (such as in point group Cnv), look at the character of the σv (NOT the σv’). If it is 1 then the subscript on the A or B is 1 If it is  1 then the subscript on the A or B is 2 4. If there are no perpendicular C2 axes and no σv, then there will be neither subscript “1” nor “2” on the symmetry label “A” or “B”. ↳ e.g., Cn, Cnh, and S2n point groups. Assigning Symmetry Labels to Irreducible Representa0ons 5. Look at the character of the inversion opera?on, i, if there is one. If it is 1 then the subscript “g” (gerade) is given If it is  1 then the subscript “u” (ungerade) is given 6. When a dis?nc?on between representa?ons is needed (i.e., if there is more than one representa?on with the same symmetry label), then look at the character of σh. If it is 1 then the symbol is given a single prime If it is  1 then the symbol is given a double prime Interpre0ng Symmetry Labels A
B
E
T
1
2
g
u
'
" Summary Symmetric with respect to principal rota0on axis An0symmetric with respect to principal rota0on axis Doubly degenerate Triply degenerate Symmetric with respect to C2 axis ┴ Cn or σv An0symmetric with respect to C2 axis ┴ Cn or σv Symmetric with respect to i (gerade) An0symmetric with respect to i (ungerade) Symmetric with respect to σh An0symmetric with respect to σh In some situa0ons these designa0ons may be modiﬁed further. Also note that such designa?ons can be used for orbitals. Interpre0ng Symmetry Labels Example: A given molecule has D4h symmetry. The dxy orbital on the central atom of the molecule has B2g symmetry. What symmetry proper0es can we determine about the dxy orbital based on this informa0on? The ﬁrst row of the D4h character table is provided to assist you. D4h E 2 C4 C2 2 C2’ 2C2’’ i 2S4 σh 2 σv 2 σd Applica0ons of Group Theory I. Chirality and Op?cal Ac?vity A molecule is chiral if it has no symmetry opera0ons (other than E) or if it only has a proper rota0on axis. Which point groups does this include? Chiral molecules have the ability to rotate plane polarized light and are therefore termed “op0cally ac0ve” II. Molecular Vibra?ons Group theory can be used to determine the possible vibra0onal and rota0onal modes in molecules, the symmetry of those mo0ons, and whether or not those mo0ons are infra red ac0ve. (You can learn more about this in a spectroscopy course. It won’t be covered it here.) III. Group theory gives us an understanding of which atomic orbitals are permi]ed to combine to form molecular orbitals. This is the applica0on that is of interest to us and we will discuss it further when we get into molecular orbital theory. Character Tables: A Summary The Character Table for a point group provides informa0on on the symmetry proper0es of atomic/molecular func0ons (and also vibra0ons) within a molecule. A Function is symmetric wrt nfold
rotation about main axis. B Function is antisymmetric wrt
nfold rotation about mai...
View Full
Document
 Winter '09
 richardson

Click to edit the document details