Point Group Assignments and Character TablesSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Consequences of SymmetrySlide Number 22Slide Number 23Point Group Assignments and Character TablesInorganic Chemistry Chapter 1: Figure 6.9 © 2009 W.H. FreemanA Simpler ApproachInorganic Chemistry Chapter 1: Table 6.2 © 2009 W.H. FreemanLINEAR MOLECULESMolecular axis is C∞ - rotation by any arbitrary angle (360/∞)o, so infinite number of rotations. Also any plane containing axis is symmetry plane, so infinite number of planes of symmetry.Divide linear molecules into two groups:Do in fact fit into scheme - but they have an infinite number of symmetry operations.(i) No centre of symmetry, e.g.:H C NC∞No C2's perp. to main axis, but ∞ σv's containing main axis: point group C∞v(ii) Centre of symmetry, e.g.:C2O C OC2C∞σhi.e. C∞ + ∞C2's + σhPoint group D∞hA few geometries have several, equivalent, highest order axes. Two geometries most important:Highly symmetrical moleculesPOINT GROUPSASSIGNMENT OF MOLECULES TO POINT GROUPSSTEP 1 : LOOK FOR AN AXIS OF SYMMETRYIf one is found - go to STEP 2If not: look for (a) plane of symmetry - if one is found, molecule belongs to point group CsA collection of symmetry operations all of which pass through a single point A point group for a molecule is a quantitative measure of the symmetry of that moleculeRegular tetrahedrone.g.ClSiClClCl4 C3 axes (one along each bond)3 C2 axes (bisecting pairs of bonds)3 S4 axes (coincident with C2's)6 σd's (each containing Si and 2 Cl's)Point group: TdRegular octahedrone.g. SFF FFFF3C4's (along F-S-F axes)also 4 C3's. 6 C2's, several planes, S4, S6 axes, and a centre of symmetry (at S atom)Point group OhThese molecules can be identified without going through the usual steps.Note: many of the more symmetrical molecules possess many more symmetry operations than are needed to assign the point group.So, What IS a group? And, What is a Character??? Character TablesInorganic Chemistry Chapter 1: Table 6.4 © 2009 W.H. FreemanInorganic Chemistry Chapter 1: Table 6.3 © 2009 W.H. FreemanInorganic Chemistry Chapter 1: Table 6.4 © 2009 W.H. FreemanInorganic Chemistry Chapter 1: Table 6.5 © 2009 W.H. FreemanInorganic Chemistry Chapter 1: Figure 6.13 © 2009 W.H. FreemanConsequences of Symmetry • Only the molecules which belong to the Cn, Cnv, or Cs group can have a permanent dipole moment. • A molecule may be chiral only if it does not have an axis of improper rotation Sn. • IR Allowed transitions may be predicted by symmetry operations • Orbital overlap may be predicted and described by symmetryCharacter table for C∞v point group E 2C∞ ... ∞ &sigmav linear, rotations quadratic A1=Σ+ 1 1 ... 1 z x2+y2, z2 A2=Σ- 1 1 ... -1 Rz E1=Π 2 2cos(Φ) ... 0 (x, y) (Rx, Ry) (xz, yz) E2=Δ 2 2cos(2φ) ... 0 (x2-y2, xy) E3=Φ 2 2cos(3φ) ... 0 ... ... ... ... ...E 2C∞ ... ∞σv i 2S∞ ... ∞C'2 linear functions, rotations quadratic A1g=Σ+g 1 1 ... 1 1 1 ... 1 x2+y2, z2 A2g=Σ-g 1 1 ... -1 1 1 ... -1 Rz E1g=Πg 2 2cos(φ) ... 0 2 -2cos(φ) ... 0 (Rx, Ry) (xz, yz) E2g=Δg 2 2cos(2φ) ... 0 2 2cos(2φ) ... 0 (x2-y2, xy) E3g=Φg 2 2cos(3φ) ... 0 2 -2cos(3φ) ... 0 ... ... ... ... ... ... ... ... ... A1u=Σ+u 1 1 ... 1 -1 -1 ... -1 z A2u=Σ-u 1 1 ... -1 -1 -1 ... 1 E1u=Πu 2 2cos(φ) ... 0 -2 2cos(φ) ... 0 (x, y) E2u=Δu 2 2cos(2φ) ... 0 -2 -2cos(2φ) ... 0 E3u=Φu 2 2cos(3φ) ... 0 -2 2cos(3φ) ... 0 ... ... ... ... ... ... ... ... ... Character table for D∞h point