Notes pertinent to lecture on Feb. 10 and 12
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1 Notes pertinent to lecture on eb. 10 and 12
2 MOLEULAR SYMMETRY Know intuitively what "symmetry" means - how to make it quantitative? Will stick to isolated, finite molecules (not crystals). SYMMETRY OPERATION arry out some operation on a molecule (or other object) - e.g. rotation. If final configuration is INDISTINGUISABLE 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.
3 e.g. for a square piece of card, rotate by 90º as shown below: o rotation 3 2 i.e. the operation of rotating by 90 o is a symmetry operation for this object Labels show final configuration is NOT identical to original. urther 90º rotations give other indistinguishable configurations - until after 4 (360º) the result is identical.
4 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.
5 Summary of symmetry elements and operations: Symmetry element Symmetry operation(s) E (identity) 1 n-1 n (rotation axis) n...n (rotation about axis) σ (reflection plane) σ (reflection in plane) i (centre of symm.) i (inversion at centre) 1 n-1 S n (rot./reflection axis)s n...sn (n even) (rot./reflection about axis) 1 2n-1 S n...sn (n odd)
6 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 O SYMMETRY Some examples for different types of molecule: e.g. rotate 2 O O O 180 o (1) (2) (2) (1) Line in molecular plane, bisecting O angle is a rotation axis, giving indistinguishable configuration on rotation by 180 o.
7 B 3 By VSEPR - trigonal, planar, all bonds equal, all angles 120 o. Take as axis a line perpendicular to molecular plane, passing through B atom. (3) (1) B 120 o B (1) (2) (2) (3) axis perpendicular to plane view down here N.B. all rotations LOKWISE when viewed along -z direction. (1) B z (3) (2)
8 Symbol for axes of symmetry n where rotation about axis gives indistinguishable configuration every (360/n) o (i.e. an n-fold axis) Thus 2 O has a 2 (two-fold) axis, B 3 a 3 (three-fold) axis. One axis can give rise to >1 rotation, e.g. for B 3, what if we rotate by 240 o? (3) (2) B 240 o B (1) (2) (3) (1) Must differentiate between two operations. Rotation by 120 o described as 3 1, rotation by 240 o as 3 2.
9 In general n axis (minimum angle of rotation (360/n) o ) gives operations n m, where both m and n are integers. When m = n we have a special case, which introduces a new type of symmetry operation... IDENTITY OPERATION or 2 O, 2 2 and for B3 3 3 both bring the molecule to an IDENTIAL arrangement to initial one. Rotation by 360 o is exactly equivalent to rotation by 0 o, i.e. the operation of doing NOTING to the molecule.
10 MORE ROTATION AXES xenon tetrafluoride, Xe 4 4 (1) (4) (4) Xe (2) 90 o (3) Xe (1) (3) (2) cyclopentadienide ion, 5 5 (1) (5) (5). (2) (4) (1) (4) 5 (3) 72 o (3) (2)
11 benzene, 6 6 (1) (6) (6) (5). (2) (3) 60 o (5) (4) (1) (2) 6 (4) (3) Examples also known of 7 and 8 axes.
12 If a 2n axis (i.e. even order) present, then n must also be present: 4 (1) (4) (4) Xe (3) (2) (3) 90 o 1 i.e. 4 Xe (2) (1) 2 i.e. 4 1 ( 2 ) 180 o (2) (3) Xe (4) (1) Therefore there must be a 2 axis coincident with 4, and the operations generated by 4 can be written: 1 4, (2 ), 3 4, 4 4 (E) Similarly, a 6 axis is accompanied by 3 and 2, and the operations generated by 6 are: 6 1, 6 2 (3 1 ), 6 3 (2 1 ), 6 4 (3 2 ), 6 5, 6 6 (E)
13 Molecules can possess several distinct axes, e.g. B 3 : 2 3 B 2 2 Three 2 axes, one along each B- bond, perpendicular to 3
14 Operation = reflection Element = plane of symmetry symbol σ Greek letter sigma Several different types of symmetry plane - different orientations with respect to symmetry axes. By convention - highest order rotation axis drawn VERTIAL. Therefore any plane containing this axis is a VERTIAL PLANE, σ v. e.g. 2 O plane above (often also called σ(xz)) an be >1 vertical plane, e.g. for 2 O there is also:
15 z O y σ(yz) - reflection leaves all atoms unshifted, therefore symmetry plane x (1) (2) This is also a vertical plane, but symmetrically different from other, could be labelled σ v '. Any symmetry plane PERPENDIULAR to main axis is a ORIZONTAL PLANE, σ h. e.g. for Xe 4 : Xe 4 Plane of molecule (perp. to 4 ) is a symmetry plane, i.e. σ h )
16 Some molecules possess additional planes, as well as σ v and σ h, which need a separate label. e.g. Xe4 Xe σ v σ d σ d σ v our "vertical" planes - but two different from others.those along bonds called σ v, but those bisecting bonds σ d - i.e. DIEDRAL PLANES Usually, but not always, σ v and σ d differentiated in same way. Two final points about planes of symmetry: (i) if no n 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) = σ 2 = E
17 INVERSION : ENTRES O SYMMETRY Best described in terms of cartesian axes: z Involves BOT rotation AND reflection. OPERATION : INVERSION ELEMENT : a POINT - ENTRE O SYMMETRY or INVERSION ENTRE. (-x,.-y, -z) inversion z x. (x, y, z) y x y 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 ENTRE O SYMMETRY (INVERSION) (i.e. ENTROSYMMETRI)
18 e.g. trans-n 2 2 (1)N. N(2) (2) inversion (2)N N(1) (1) (1) centre of symmetry (2) In practice, inversion involves taking every atom to the centre - and out the same distance in the same direction on the other side. Symbol - same for operation (inversion) and element (centre): Another example: Xe 4 (1) (3) i (4) Xe (2) i (2) Xe (4) (3) Xe atom is centre of symmetry (1)
19 As for reflections, the presence of a centre of symmetry only generates one new operation, since carrying out inversion twice returns everything back to start. (x, y, z) i (-x, -y, -z) i (x, y, z) i.e. (i)(i) = i 2 = E Inversion is a OMPOSITE operation, with both rotation and reflection components. onsider a rotation by 180 o about the z axis: (x, y, z) (-x, -y, z) ollow this by reflection in the xy plane (-x, -y, z) (-x, -y, -z)
20 BUT individual components need not be symmetry operations themselves... e.g. staggered conformation of lbr-lbr. Br l Inversion at centre gives indistinguishable configuration. l Br centre The components, of rotation by 180 o or reflection in a plane perpendicular to the axis, do not. If, however, a molecule does possess a 2 axis and a σ h (perpendicular) plane as symmetry operations, then inversion (i) must also be a symmetry operation.
21 IMPROPER ROTATIONS : ROTATION-RELETION AXES Operation: clockwise rotation (viewed along -z direction) followed by reflection in a plane perpendicular to that axis. Element: rotation-reflection axis (sometimes known as "alternating axis of symmetry") As for inversion - components need not be themselves symmetry operations for the molecule.
22 e.g. a regular tetrahedral molecule, such as 4 rotate 90 o reflect four-fold rotation reflection S 4 1 Symbols: rotation-reflection axis S n (element) rotation-reflection S n m (operations) where rotation is through (360/n) o
23 S 4 axis requires presence of coincident 2 axis If n and σ h are both present individually - there must also be an S n axis : e.g. B 3 - trigonal planar B 3, S 3 σ h in plane of molecule σh individually, 1 therefore S 3 must also be a symmetry operation Other S n examples: I 7, pentagonal bipyramid, has 5 and σ h, therefore S 5 also. Ethane in staggered conformation S 6 i.e. rotate by 60 o and reflect in perp. plane. Note NO 6, σ h separately.
24 POINT GROUPS A 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 O MOLEULES TO POINT GROUPS STEP 1 : LOOK OR AN AXIS O SYMMETRY If one is found - go to STEP 2 If not: look for (a) plane of symmetry - if one is found, molecule belongs to point group s
25 POINT GROUPS A 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 O MOLEULES TO POINT GROUPS STEP 1 : LOOK OR AN AXIS O SYMMETRY If one is found - go to STEP 2 If not: look for (a) plane of symmetry - if one is found, molecule belongs to point group s
26 e.g. SOl 2 O If no plane is found, look for.. No axis, but plane containing S, S O, bisecting lsl angle, is a l symmetry plane. ence s l point group. (b) centre of symmetry - if one is found, molecule belongs to point group i. e.g. lbrlbr (staggered conformation): l Br Br l No axis, no planes, but mid-point of - bond is centre of symmetry. Therefore i point group. No axes, plane or centre, therefore (so called because E = 1, (c) no symmetry except E : point group 1 rotation through 360 o ) e.g. lbr Br l No symmetry except E, therefore point group 1.
27 STEP 2 : LOOK OR 2 AXES PERPENDIULAR TO n If found, go to STEP 3. If not, look for (a) a ORIZONTAL PLANE O SYMMETRY, if found - point group is nh ( n = highest order axis) e.g. trans-n 2 2 : ighest order axis is 2 (perp. to plane, through mid-pt. of N=N bond). N. N No 2 axes perp. to this, but molecular plane is plane of symmetry (perp. to 2, i.e. σ h ). Point group 2h. If there is no horizontal plane, look for (b) n VERTIAL PLANES O SYMMETRY. If found, molecule belongs to point group nv
28 Many examples, e.g. 2 O 2 axis as shown. No other 2 's, no O 2 σ h, but two sv's, one in plane, one perp. to plane, bisecting O angle. Point group 2v Pl 3 3 highest order axis l P 3 l l No 2 's perp. to 3 No σ h, but 3 σ v 's, each contains P, one l Br 5 By VSEPR, square pyramidal Therefore 3v ighest order axis : 4 No 2 's perp. to 4 4 Br No σ h, but 4 vertical planes. Therefore 4v
29 N.B. of 4 vertical planes, two are σ v 's, two σ d 's σ v σ d Br σ v (looking down 4 axis) σ d If no planes at all, could have (c) no other symmetry elements: point group n, or (d) an S 2n axis coincident with n : point group S 2n
30 STEP 3 If there are n 2 's perp. to n, look for: (a) horizontal plane of symmetry. If present, point group is D nh e.g. ethene (ethylene), ighest order axis 2 - along = bond Two additional 2 's as shown. σ h in plane defined by the last two 2 's Point group D 2h B 3 Planar trigonal molecule by VSEPR B 3 2 Main axis 's perp. to 3 Point group D 3h 2 2 σ h - plane of molecule
31 Xe 4 Square planar by VSEPR 2 " Xe 4 2 ' Main axis 's perp. to 4 (2 along Xe bonds ( 2 '), 2 bisecting, ( 2 ")) σ h - plane of molecule 2 " 2 ' Point group D 4h If no σ h, look for: (b) n vertical planes of symmetry (σ v /σ d ). If these are present, molecule belongs to point group D nd
32 e.g. allene, 2 == 2. 2 (main axis) 2 ' Looking down == bond 2 ' Main axis 2 - along == Two 2 's as shown Two vertical planes (σ d ) - each containing one 2 unit Point group D 2d A few molecules do not appear to fit into this general scheme...
33 LINEAR MOLEULES Do in fact fit into scheme - but they have an infinite number of symmetry operations. Molecular axis is - 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: (i) No centre of symmetry, e.g.: N No 2 's perp. to main axis, but σ v 's containing main axis: point group v
34 (ii) entre of symmetry, e.g.: 2 O O σ h 2 i.e. + 2 's + σ h Point group D h ighly symmetrical molecules A few geometries have several, equivalent, highest order axes. Two geometries most important:
35 Regular tetrahedron e.g. l 4 3 axes (one along each bond) 3 2 axes (bisecting pairs of bonds) l Si l l 3 S 4 axes (coincident with 2 's) 6 σ d 's (each containing Si and 2 l's) Point group: T d e.g. Regular octahedron S 3 4 's (along -S- axes) also 4 3 's. 6 2 's, several planes, S 4, S 6 axes, and a centre of symmetry (at S atom) Point group O h These 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.
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