MOLECULAR SYMMETRY

1 MOLECULAR 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) - 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 Indistinguishable does not necessarily mean identical . for a square piece of card, rotate by 90 as shown below: 1 2 4 34 1 3 . the opera tion of rotating by 90o is a s ymmetry operation for this objectLabels show final configuration is NOT identical to original.

2 Further 90 rotations give other indistinguishable configurations - until after 4 (360 ) 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 opera tion is performed. Example a bove - operation was rotation, ele ment was a ROTATION AXIS. Other examples la of SYMMETRY elements and operations: SYMMETRY element SYMMETRY opera tion(s) E (identity)Cn (rotation axis ) (rotation about axis ) (reflection plane) (reflection in plane)i (centre of symm.)

3 I (inversion at centre)Sn ( axis ) (n even) ( about axis ) (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 - AXES OF SYMMETRYSome e xamples for different types of molecule: H2OO(1)HH(2)O(2)HH(1)rotate180oLine in MOLECULAR plane, bisecting HOH angle is a rotation axis , giving indistinguishable c onfiguration on rotation by VSEPR - trigonal, planar, all bonds e qual, all angles 1 20o. Take a s axis a line perpendicular to MOLECULAR plane, passing through B (1)FF(2)F(3)B(2)FF(3)F(1)12 0oax is perpendicularto all rotations CLOCKWISE when vie wed along -z direction.

4 (1)FBF(2)F(3)zvie w down hereSymbol for axes of symmetryCnwhere rotation about axis gives indistinguishable configuration eve ry (360 /n)o ( a n n-fold axis)Thus H2O has a C2 (two-fold) axis , BF3 a C3 (three-fold) axis. One axis c an give rise to >1 rotation, for BF3, what if we rotate by 240o?B(1)FF(2)F(3)B(3)FF(1)F(2)240oMust differentiate between two by 120o described as C31, rotation by 240o as general Cn a xis (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 IDENTITY OPERATIONFor H2O, C22 a nd for BF3 C33 both bring the molecule to an IDENTICAL arrangement to initial by 360o is e xactly e quivale nt to rotation by 0o.

5 The operation of doing NOTHING to the ROTATION AXES xe non tetrafluoride, XeF4C4Xe(4)FF(1)F(3)F(2)Xe(3)FF(4)F(2)F( 1)90ocyclopentadienide ion, C5H5 CCCCCH(1)H(2)H(3)(4)H(5)HC5 CCCCCH(5)H(1)H(2)(3)H(4) , C6H6 CCCCCCH(1)H(2)H(3)H(4)(5)H(6)HC6 CCCCCCH(6)H(1)H(2)H(3)(4)H(5) also known of C7 a nd C8 a xes. If a C2n axis ( 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) 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 a xis is accompanied by C3 a nd C2, a nd the operations generated by C6 a re :C61, C62 (C31), C63 (C21), C64 (C32), C65, C66 (E)Mole cules can posses s several distinct axes , e.

6 G. BF3:C3 FBFFC2C2C2 Three C2 a xe s, one along eac h B-F bond, perpendicular to C3 Operation = reflectionElement = plane of symmetrysy mbol Greek letter sigma Seve ral differe nt types of SYMMETRY plane -different orientations with respect to SYMMETRY axe convention - highest order rotation axis drawn VERTICAL. Therefore any plane containing this axis is a VERTICAL PLANE, H2O plane above (often also called (xz))Can be >1 vertical plane, for H2O there is also:H(2)O(1)Hzyx (yz) - reflection le ave s all atoms unshifted, therefore sy mmetry planeThis is also a vertical plane, but symmetrically different from other, could be labelled v'.Any SYMMETRY plane PERPENDICULAR to main axis is a HORIZONTAL PLANE, h.

7 For XeF4:C4 XeFFFFP lane of molecule (perp. to C4) is a SYMMETRY plane, h)Some molecules posse ss a dditional planes, a s well as v a nd h, which need a s epara te label. XeF4 FXeFFF v v d dFour "ve rtical" planes - but two differe nt from others .Those a long bonds calle d v, but those bisecting bonds d - DIHEDRAL PLANESU sually, but not always, v a nd d differentiated in same final points a bout planes of SYMMETRY :(i) if no Cn a xis, plane just calle d ;(ii) unlike rotations, only ONE operation per plane. A se cond reflection returns y ou to original state, ( )( ) = 2 = EINVERSI ON : CENTRES OF SYMMETRY zyxzyxinversion.(x, y, z).(-x, -y, -z)The origin, (0, 0, 0) is the centre of inversion.

8 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) ( CENTROSYMMETRIC)Involves BOTH rotation AND : INVERSIONELEMENT : a POINT - CENTRE OF SYMMETRY or INVERSION described in terms of cartesian axes: trans-N2F2(1)NN(2)(1)FF(2)(2)NN(1)(2)FF( 1) ntre of symmetryIn practice, inversion involves taking every atom to the centre - and out the same distance in the same direction on the other - same for operation (invers ion) and element (centre): iAnother example: XeF4(4)FXeF(2)F(3)F(1)(2)FXeF(4)F(1)F(3) iXe a tom is c entre of symmetryAs for reflections, the presence of a c entre of SYMMETRY only generates one new operation, since c arrying out invers ion twice returns eve ry thing back to start.

9 (x, y, z)i(-x, -y, -z)i(x, y, z) (i)(i) = i2 = EInversion is a COMPOSITE opera tion, with both rotation and reflection components. Consider a rotation by 1 80o a bout the z axis:(-x, -y, z)(x, y, z)Follow this by reflection in the xy plane(-x, -y, z) (-x, -y, -z)BUT individual components need not be SYMMETRY operations themselv staggered conformation of at centre gives indistinguishable components, of rotation by 180o or re flection in a plane perpendicular to the axis , do , however, a molecule does possess a C2 a xis a nd a h (perpendicular) plane as SYMMETRY opera tions, then invers ion (i) must als o be a SYMMETRY ROTATIONS :ROTATION-REFLECTION AXESO peration: clockwise rotation (viewed along -z direction) followed by reflection in a plane perpendicular to that.

10 Rotation-reflection axis (sometimes known as "alternating axis of SYMMETRY ")As for inversion - components need not be themselves sy mmetry opera tions for the a regular tetrahedral molecule, such as CH4 HCHHHHCHHHHCHHH rotate90oreflectfour-fold rotation reflectionS41 Symbols: rotation-reflection axis Sn (element) rotation-reflection Snm (operations)where rotation is through (360/n)oS4 a xis requires prese nce of coincident C2 a xisIf Cn and h are both present individually - there must also be an Sn axis BF3 - trigonal planarFBFFC3, S3 h in plane of molecule. C31 + h individually, therefore S31 must also be a SYMMETRY operationOther Sn examples: IF7, pentagonal bipyramid, has C5 and h, therefore S5 in staggered.