Notes pertinent to lecture on Feb. 10 and 12Slide 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 20Slide 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 35Notes pertinent to lecture on Feb. 10 and 12MOLECULAR SYMMETRYKnow intuitively what "symmetry" means - how to make it quantitative?Will stick to isolated, finite molecules (not crystals). SYMMETRY OPERATION Carry out some operation on a molecule (or other object) - e.g. rotation. If final configuration is INDISTINGUISHABLE from the initial one - then the operation is a SYMMETRY OPERATION for that object. The line, point, or plane about which the operation occurs is a SYMMETRY ELEMENT N.B. “Indistinguishable” does not necessarily mean “identical”.e.g. for a square piece of card, rotate by 90º as shown below: 1 2 4 34 1 3 290orotationi.e. the operation of rotating by 90o is a symmetry operation for this objectLabels show final configuration is NOT identical to original. Further 90º rotations give other indistinguishable configurations - until after 4 (360º) the result is identical.SYMMETRY OPERATIONS Motions of molecule (rotations, reflections, inversions etc. - see below) which convert molecule into configuration indistinguishable from original. SYMMETRY ELEMENTS Each element is a LINE, PLANE or POINT about which the symmetry operation is performed. Example above - operation was rotation, element was a ROTATION AXIS. Other examples later.Summary of symmetry elements and operations: Symmetry element Symmetry operation(s) – E (identity)Cn (rotation axis) Cn1.....Cnn-1 (rotation about axis)σ (reflection plane) σ (reflection in plane)i (centre of symm.) i (inversion at centre)Sn (rot./reflection axis)Sn1...Snn-1 (n even) (rot./reflection about axis) Sn1...Sn2n-1 (n odd)Notes(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 C3Operation = reflectionElement = plane of symmetrysymbol σGreek letter ‘sigma’ Several different types of symmetry plane -different orientations with respect to symmetry axes.By convention - highest order rotation axis drawn VERTICAL. Therefore any plane containing this axis is a VERTICAL PLANE, σv.e.g. H2O plane above (often also called σ(xz))Can be >1 vertical plane, e.g. for H2O there is also:H(2)O(1)Hzyxσ(yz) - reflection leaves all atoms unshifted, therefore symmetry planeThis is also a vertical plane, but symmetrically different from other, could be labelled σv'.Any symmetry plane PERPENDICULAR to main axis is a HORIZONTAL PLANE, σh. e.g. for XeF4:C4XeFF FFPlane of molecule (perp. to C4) is a symmetry plane, i.e. σh)Some molecules possess additional planes, as well as σv and σh, which need a separate label. e.g. XeF4FXe FFFσvσvσdσdFour "vertical" planes - but two different from others.Those along bonds called σv, but those bisecting bonds σd - i.e. DIHEDRAL PLANESUsually, but not always, σv and σd differentiated in same way.Two final points about planes of symmetry:(i) if no Cn axis, plane just called σ;(ii) unlike rotations, only ONE operation per plane. A second reflection returns you to original state, i.e. (σ)(σ) = σ2 = EINVERSION : CENTRES OF SYMMETRYzyxzyxinversion.(x, y, z).(-x, -y, -z)The origin, (0, 0, 0) is the centre of inversion. If the coordinates of every point are changed from (x,y,z) to (-x, -y, -z), and the resulting arrangement is indistinguishable from original - the INVERSION is a symmetry operation, and the molecule possesses a CENTRE OF SYMMETRY (INVERSION) (i.e. CENTROSYMMETRIC)Involves BOTH rotation AND reflection.OPERATION : INVERSIONELEMENT : a POINT - CENTRE OF SYMMETRY or INVERSION CENTRE.Best described in terms of