SymmetrySymmetry Operations and Symmetry ElementsSlide 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 18Consequences of SymmetryPoint Group Assignments and Character TablesSlide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Symmetry Intuitively, we know symmetry when we see it. But how do we put in quantitative terms that allows us to compare, assign, classify?Symmetry Operations and Symmetry Elements Definitions:  A symmetry operation is an operation on a body such that, after the operation has been carried out, the result is indistinguishable from the original body (every point of the body is coincident with an equivalent point or the same point of the body in its original orientation).  A symmetry element is a geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out Symmetry Operation Symmetry Element Notation Identity - E Reflection in a plane Plane of symmetry σ Proper rotation Rotation axis (line) Cn Rotation followed by reflection in the plane perpendicular to the rotation axis Improper rotation axis (line) Sn Inversion Center of inversion I σv, σd, σh Cn ; where = 360/angleNotes(i) symmetry operations more fundamental, but elements often easier to spot.(ii) some symmetry elements give rise to more than one operation - especially rotation - as above.ROTATIONS - AXES OF SYMMETRYSome examples for different types of molecule: e.g. H2OO(1)H H(2)O(2)H H(1)rotate180oLine in molecular plane, bisecting HOH angle is a rotation axis, giving indistinguishable configuration on rotation by 180o.BF3By VSEPR - trigonal, planar, all bonds equal, all angles 120o. Take as axis a line perpendicular to molecular plane, passing through B atom.B(1)F F(2)F(3)B(2)F F(3)F(1)120oaxis perpendicularto planeN.B. all rotations CLOCKWISE when viewed along -z direction.(1)F BF(2)F(3)zview down hereSymbol for axes of symmetryCnwhere rotation about axis gives indistinguishable configuration every (360/n)o (i.e. an n-fold axis)Thus H2O has a C2 (two-fold) axis, BF3 a C3 (three-fold) axis. One axis can give rise to >1 rotation, e.g. for BF3, what if we rotate by 240o?B(1)F F(2)F(3)B(3)F F(1)F(2)240oMust differentiate between two operations.Rotation by 120o described as C31, rotation by 240o as C32.In general Cn axis (minimum angle of rotation (360/n)o) gives operations Cnm, where both m and n are integers.When m = n we have a special case, which introduces a new type of symmetry operation..... IDENTITY OPERATIONFor H2O, C22 and for BF3 C33 both bring the molecule to an IDENTICAL arrangement to initial one.Rotation by 360o is exactly equivalent to rotation by 0o, i.e. the operation of doing NOTHING to the molecule.MORE ROTATION AXES xenon tetrafluoride, XeF4C4Xe(4)FF(1)F(3)F(2)Xe(3)FF(4)F(2)F(1)90ocyclopentadienide ion, C5H5–CC CCCH(1)H(2)H(3)(4)H(5)HC5CC CCCH(5)H(1)H(2)(3)H(4)H.72obenzene, C6H6CCCCCCH(1)H(2)H(3)H(4)(5)H(6)HC6CCCCCCH(6)H(1)H(2)H(3)(4)H(5)H60o.Examples also known of C7 and C8 axes.If a C2n axis (i.e. even order) present, then Cn must also be present:C4Xe(4)FF(1)F(3)F(2)Xe(3)FF(4)F(2)Xe(2)FF(1)F(3)F(1)F(4)90oi.e. C41180oi.e. C42 (≡ C21)Therefore there must be a C2 axis coincident with C4, and the operations generated by C4 can be written:C41, C42 (C21), C43, C44 (E)Similarly, a C6 axis is accompanied by C3 and C2, and the operations generated by C6 are:C61, C62 (C31), C63 (C21), C64 (C32), C65, C66 (E)Molecules can possess several distinct axes, e.g. BF3:C3FBF FC2C2C2Three C2 axes, one along each B-F bond, perpendicular to C3X ,Y, Z -X, -Y, -ZSo, What IS a group? And, What is a Character??? Symmetry elements/operations can be manipulated by Group Theory, Representations and Character Tables Character TablesInorganic Chemistry Chapter 1: Table 6.4 © 2009 W.H. FreemanInorganic Chemistry Chapter 1: Table 6.3 © 2009 W.H. FreemanConsequences of Symmetry • Only the molecules which belong to the Cn, Cnv, or Cs point 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 symmetryPoint Group Assignments and Character TablesPOINT GROUPSA 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 molecule Assignment of Symmetry Elements to Point Group: At first Looks Daunting.Inorganic Chemistry Chapter 1: Figure 6.9 © 2009 W.H. Freeman Daunting? However almost all we will be concerned with belong to just a few symmetry point groupsA Simpler ApproachPOINT 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 moleculePoint Group Assignments: MFT Ch. 4LINEAR 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 moleculesRegular 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