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Part II: Symmetry Operations and Point Groups

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Part II: Symmetry Operations and Point Groups Part II: Symmetry Operations and Point Groups C734b C734b Symmetry Operations and 1 Point groups Definitions 1.- symmetry operations: leave a set of objects in indistinguishable configurations said to be equivalent -The identity operator, E is the “do nothing” operator. Therefore, its final configuration is not distinguishable from the initial one, but identical with it. 2.- symmetry element: a geometrical entity (line, plane or point) with respect to which one or more symmetry operations may be carried out. Four kinds of symmetry elements for molecular symmetry 1.) Plane operation = reflection in the plane 2.) Centre of symmetry or inversion centre: operation = inversion of all atoms through the centre 3.) Proper axis operation = one or more rotations about the axis 4.) Improper axis operation = one or more of the sequence rotation about the axis followed by reflection in a plane perpendicular (┴) to the rotation axis. C734b Symmetry Operations and 2 Point groups 1 1. Symmetry Plane and Reflection A plane must pass through a body, not be outside. Symbol = σ. The same symbol is used for the operation of reflecting through a plane σm as an operation means “carry out the reflection in a plane normal to m”. Take a point {e , e , e } along (xˆ, yˆ,zˆ) ∴ 1 2 3 ⎧ − ⎫ σ {e , e , e } = {-e , e , e } ≡ ⎨e1 , e 2 , e 3 ⎬ x 1 2 3 1 2 3 ⎩ ⎭ Often the plane itself is specified rather than the normal. σ = σ means “reflect in a plane containing the y- and z-, usually ⇒ x yz called the yz olane C734b Symmetry Operations and 3 Point groups -atoms lying in a plane is a special case since reflection through a plane doesn’t move the atoms. Consequently all planar molecules have at least one plane of symmetry ≡ molecular plane Note: σ produces an equivalent configuration. σ2 = σσ produces an identical configuration with the original. ∴ σ2 = E ∴ σn = E for n even; n = 2, 4, 6,… σn = σ for n odd; n = 3, 5, 7,… C734b Symmetry Operations and 4 Point groups 2 Some molecules have no σ-planes: S O F Cl Linear molecules have an infinite number of planes containing the bond axis. X Y Many molecules have a number of planes which lie somewhere between these two extremes: Example: H O 2 planes; 1 molecular plane + 2 O the other bisecting the H-O-H group H H C734b Symmetry Operations and 5 Point groups Example: NH3 N 3 planes containing on N-H bond and H H bisecting opposite HNH group H 4 planes; I molecular plane + Example: BCl3 Cl Cl 3 containing a B-Cl bond and B bisecting the opposite Cl-B-Cl group Cl - 5 planes; 1 molecular plane + Example: [AuCl ]- 4 Cl Cl 4 planes; 2 containing Cl-Au-Cl square planar Au + 2 bisecting the Cl-Au-Cl groups. Cl Cl C734b Symmetry Operations and 6 Point groups 3 Tetrahedral molecules like CH4 have 6 planes. H C H H H Octahedral molecules like SF6 have 9 planes in total F F F S F F F C734b Symmetry Operations and 7 Point groups 2.) Inversion Centre -Symbol = i - operation on a point {e1, e2, e3} i{e1, e2, e3} = {-e1, -e2, -e3} z z i e -e 3 1 -e 2 e y 2 y e -e 1 3 x x C734b Symmetry Operations and 8 Point groups 4 -If an atom exists at the inversion centre it is the only atom which will not move upon inversion. -All other atoms occur in pairs which are “twins”. This means no inversion centre for molecules containing an odd number of more than one species of atoms. i2 = ii = E ⇒ in = E n even in = i n odd C734b Symmetry Operations and 9 Point groups Example: 2 1 4 - 3 - Cl Cl Cl Cl Au i Au Cl Cl Cl Cl 3 4 1 2 - but H H or H H C H H H H H have no inversion centre even though in the methane case the number of Hs is even C734b Symmetry Operations and 10 Point groups 5 3. Proper Axes and Proper Rotations - A proper rotation or simply rotation is effected by an operator R(Φ,n) which means “carry out a rotation with respect to a fixed axis through an angle Φ described by some unit vector n”. ’ For example: R(π/4, x){e1, e2, e3} = {e1’, e2 , e3’} z z R(π/4, x) 45o e ' e 2 3 e ' 45o 3 e y 2 y e 1 e ' 1 x x C734b Symmetry Operations and 11 Point groups Take the following as the convention: a rotation is positive if looking down axis of rotation the rotation appears to be counterclockwise. More common symbol for rotation operator is Cn where n is the order of the axis. C means “carry out a rotation through an angle of Φ = 2π/n” ⇒ n R(π/4) ≡ C R(π/2) = C R(π) = C etc. ∴ 8 4 2 C2 is also called a binary rotation. C734b Symmetry Operations and 12 Point groups 6 Product of symmetry operators means: “carry out the operation successively beginning with the one on the right”. C C = C 2 = C = R(π, n) ∴ 4 4 4 2 k k (Cn) = Cn = R(Φ, n); Φ = 2πk/n C -k = R(-Φ, n); Φ = -2πk/n ∴ n k -k k+(-k) 0 Cn Cn = Cn = Cn ≡ E C -k is the inverse of C k ⇒ n n C734b Symmetry Operations and 13 Point groups Example: 3 2 1 C3 ≡ C3 = rotation by 2π/3 2 1 1 3 I = 120o II C3 axis is perpendicular to the plane of the equilateral triangle. 2 1 2 C3C3 = C3 = rotation by 4π/3 3 2 1 3 = 240o I III C734b Symmetry Operations and 14 Point groups 7 But 1 2 -1 C3 = rotation by -2π/3 3 2 1 3 III I = -120o C 2 = C -1 ∴ 3 3 2 2 3 C3 = rotation by 2π 3 1 3 1 I I = 360o C 3 = E ∴ 3 C734b Symmetry Operations and 15 Point groups 4 What about C3 ? 3 2 4 C3 = C3C3C3C3 2 1 1 II I 3 C 4 = C ∴ 3 3 only C , C 2, E are separate and distinct operations ⇒ 3 3 Similar arguments can be applied to any proper axis of order n C734b Symmetry Operations and 16 Point groups 8 Example: C6: C6; 2 C6 ≡ C3 3 C6 ≡ C2 4 -2 -1 C6 ≡ C6 ≡ C3 5 -1 C6 ≡ C6 6 C6 ≡ E Note: for Cn n odd the existence of one C2 axis perpendicular to or σ containing Cn implies n-1 more separate (that is, distinct) C2 axes or σ planes 1 C5 axis coming out of the page - 2 H 5 H H H H 3 4 C734b Symmetry Operations and 17 Point groups * When > one symmetry axis exist, the one with the largest value of n ≡ PRINCIPLE AXIS Things are a bit more subtle for Cn, n even Take for example C4 axis: C2(1) 2 1 C2(2) 3 4 C734b Symmetry Operations and 18 Point groups 9 C2(2) 1 4 C4 C2(1) 2 3 i.e.; C2(1) → C2(2); C2(2) → C2(1) C2(1) 4 3 C4 C2(2) 2 Total = C4 = C2 i.e.; C2(1) → C2(1); C2(2) → C2(2) 1 2 C734b Symmetry Operations and 19 Point groups C2(1) 2 1 C4 C2(2) 3 -1 Total C4 = C4 3 4 i.e.; C2(1) → C2(2); C2(2) → C2(1) Conclusion: C2(1) and C2(2) are not distinct Conclusion: a Cn axis, n even, may be accompanied by n/2 sets of 2 C2 axes C734b Symmetry Operations and 20 Point groups 10 b 2 a2 b1 a1 4 C2 axes: (a1, a2) and (b1, b2). Cn rotational groups are Abelian C734b Symmetry Operations and 21 Point groups 4. Improper Axes and Improper Rotations Accurate definitions: Improper rotation is a proper rotation R(Φ, n) followed by inversion ≡ iR(Φ, n) Rotoreflection is a proper reflection R(Φ, n) followed by reflection in a plane normal to the axis of rotation, σh Called Sn = σhR(Φ, n) Φ = 2π/n Cotton and many other books for chemists call Sn an improper rotation, and we will too. C734b Symmetry Operations and 22 Point groups 11 Example: staggered ethane (C3 axis but no C6 axis) 1 4 1 2 3 2 σh 6 5 C 6 3 4 6 4 1 2 6 5 5 3 C734b Symmetry Operations and 23 Point groups Note: 1 4 6 4 3 2 C6 6 5 σ 6 h 5 6 1 4 1 2 3 2 5 3 S = σ C or C σ ⇒ n h n n h The order is irrelevant. C734b Symmetry Operations and 24 Point groups 12 Clear if Cn exists and σh exists Sn must exist. HOWEVER: Sn can exist if Cn and σh do not. The example above for staggered ethane is such a case. 2 3 The element Sn generates operations Sn, Sn , Sn , … However the set of operations generated are different depending if n is even or odd. n even {S , S 2, S 3, …, S n} 2 2 3 3 n n n n n n ≡ {σhCn, σh Cn , σh Cn , …, σh Cn } n But σ n = E and C n = E Sn = E h n ⇒ n+1 n+2 2 m m and therefore: Sn = Sn, Sn = Sn , etc, and Sn = Cn if m is even.
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