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.

C734b Symmetry Operations and 25 Point groups

Therefore for S6, operations are:

S6 2 2 S6 ≡ C6 ≡ C3 3 S6 ≡ S2 ≡ i 4 4 2 S6 ≡ C6 ≡ C3 5 -1 S6 ≡ S6 6 S6 ≡ E

Conclusion: the existence of an Sn axis requires the existence of a Cn/2 axis.

Sn groups, n even, are Abelian.

C734b Symmetry Operations and 26 Point groups

13 n odd n n n Consider Sn = σh Cn =σhE= σh

⇒ σh must exist as an element in its own right, as must Cn.

Consider as an example an S5 axis. This generates the following operations:

S 1 = σ C 5 h 5 6 6 6 S5 = σh C5 ≡ C5 S 2 = σ 2C 2 ≡ C 2 5 h 5 5 7 7 7 2 S5 = σh C5 ≡σhC5 S 3 = σ 3C 3 ≡σC 3 5 h 5 h 5 8 8 8 3 S5 = σh C5 ≡ C5 S 4 = σ 4C 4 ≡ C 4 5 h 5 5 9 9 9 4 S5 = σh C5 ≡σhC5 S 5 = σ 5C 5 ≡σ 5 h 5 h 10 10 10 S5 = σh C5 ≡ E

11 The element S , n odd, generates 2n It’s easy to show that S5 = S5 ∴ n distinct operations Sn groups, n odd, are not Abelian

C734b Symmetry Operations and 27 Point groups

a) A body or molecule for which the only symmetry operator is E has no symmetry at all. However, E is equivalent to a rotation through an angle Φ = 0 about an arbitrary axis. It is not customary to include C1 in a list of symmetry elements except when the only symmetry operator is the identity E. b) Φ = 2π/n; n is a unit vector along the axis of rotation.

C734b Symmetry Operations and 28 Point groups

14 Products of Symmetry Operators

-Symmetry operators are conveniently represented by means of a stereogram or stereographic projection.

Start with a circle which is a projection of the unit sphere in configuration space (usually the xy plane). Take x to be parallel with the top of the page.

A point above the plane (+z-direction) is represented by a small filled circle.

A point below the plane (-z-direction) is represented by a larger open circle.

A general point transformed by a point symmetry operation is marked by an E.

C734b Symmetry Operations and 29 Point groups

For improper axes the same geometrical symbols are used but are not filled in.

C734b Symmetry Operations and 30 Point groups

15 C734b Symmetry Operations and 31 Point groups

C734b Symmetry Operations and 32 Point groups

16 Example: Show that S+(Φ,z) = iR-(π-Φ, z)

Example: Prove iC2z = σz

C734b Symmetry Operations and 33 Point groups

The complete set of point-symmetry operators including E that are generated from the operators {R1, R2, …} that are associated with the symmetry elements {C1, i, Cn, Sn, σ} by forming all possible products like R2R1 satisfy the necessary group properties:

1) Closure 2) Contains E 3) Satisfies associativity 4) Each element has an inverse

Such groups of point symmetry operators are called Point Groups

C734b Symmetry Operations and 34 Point groups

17 Example: construct a multiplication table for the S4 point group having the set of + 2 - elements: S4 = {E, S4 , S4 = C2, S4 }

+ − S4 E S4 C2 S4 + − E E S4 C2 S4 + + S4 S4

C2 C2 − − S4 S4

+ + + - + Complete row 2 using stereograms: S4 S4 , C2S4 , S4 S4 (column x row)

C734b Symmetry Operations and 35 Point groups

C734b Symmetry Operations and 36 Point groups

18 Complete Table

+ − S4 E S4 C2 S4 + − E E S4 C2 S4 + + − S4 S4 C2 S4 E − + C2 C2 S4 E S4 − − + S4 S4 E S4 C2

C734b Symmetry Operations and 37 Point groups

Another example: an equilateral triangle

Choose C3 axis along z

+ - The set of distinct operators are G = (E, C3 , C3 , σA, σB, σC} y

b

a x c

Blue lines denote symmetry planes.

C734b Symmetry Operations and 38 Point groups

19 b

Let Ψ1 = a

c c b = Ψ σ Ψ = a 4 a = Ψ1 A 1 ∴ EΨ1 = b c a b

+ c = Ψ2 σ Ψ = c = Ψ C3 Ψ1 = B 1 5

b a c a

σCΨ1 = b = Ψ3 b = Ψ6 - C3 Ψ1= a c

C734b Symmetry Operations and 39 Point groups

Typical binary products:

c

b - C +C + = C - + + + ≡ C3 Ψ1 ⇒ 3 3 3 C3 C3 Ψ1 = C3 Ψ2 a b

+ - + C +C - = E C3 C3 Ψ1 = C3 Ψ3 a ≡ EΨ1 ⇒ 3 3

c a ≡σ Ψ + + + b C 1 ⇒ C3 σA = σC C3 σAΨ1 = C3 Ψ4

c b σ C +Ψ = C +Ψ ≡σ Ψ σ C + = σ A 3 1 3 2 c B 1 ⇒ A 3 B etc. a + Note: i) σA and C3 do not commute ii) Labels move, not the symmetry elements C734b Symmetry Operations and 40 Point groups

20 Can complete these binary products to construct the multiplication table for G

+ - Multiplication Table for the set G = {E, C3 , C3 , σA, σB, σC}

+ − G E C3 C3 σ A σ B σ C + − E E C3 C3 σ A σ B σ C + + − C3 C3 C3 E σ C σ A σ B − − + C3 C3 E C3 σ B σ C σ A + − σ A σ A σ B σ C E C3 C3 − + σ B σ B σ C σ A C3 E C3 + − σ C σ C σ A σ B C3 C3 E

C734b Symmetry Operations and 41 Point groups

Symmetry Point Groups (Schönflies notation) One symmetry element

1.) No symmetry: {E exists} order = 1 C1

2 2.) sole element is a plane: Cs {σ, σ = E} order = 2

{i, i2 = E} order = 2 3.) sole element is an inversion centre: Ci

4.) Only element is a proper axis of order n: Cn

1 2 n {Cn , Cn , …, Cn = E} order = n These are Abelian cyclic groups.

C734b Symmetry Operations and 42 Point groups

21 5.) Only element is an improper axis of order n.

Two cases:

3 n-1 a) n even {E, Sn, Cn/2, Sn , …, Sn order = n

S = C Note: S2 = i. ⇒ 2 i

Symbol: Sn

b) n odd order = 2n including σh and operations generated by Cn axis.

Symbol: Cnh

C734b Symmetry Operations and 43 Point groups

Two or more symmetry elements

Need to consider (1) the addition of different symmetry elements to a Cn axis and ’ (2) addition of symmetry planes to a Cn axis and nC2 axes perpendicular to it.

To define symbols, consider the principle axis to be vertical.

⇒ Symmetry plane perpendicular to Cn will be a horizontal plane σh

There are 2 types of vertical planes (containing Cn)

If all are equivalent ⇒ σv ( v ≡ vertical)

There may be 2different sets (or classes)

one set = σv; the other set = σd (d = dihedral)

C734b Symmetry Operations and 44 Point groups

22 ∴ adding σh to Cn → Cnh (≡ Sn; n odd)

adding σv to Cn: n odd → nσv planes n even → n/2 σv planes and n/2 σd planes

See previous discussion regarding C4 axis.

Note: the σd set bisect the dihedral angle between members of the σv set.

C point group. Distinction is arbitrary: ⇒ nv

C734b Symmetry Operations and 45 Point groups

’ Next: add σh to Cn with n C2 axes

⇒ Dnh point group (D for dihedral groups)

Note: σhσv = C2. Therefore, need only find existence of Cn, σh, and σv’s to establish Dnh group.

’ By convention however, simultaneous existence of Cn, n C2 s, and σh used as criterion.

’ ’ Next: add σd s to Cn and n C2 axes.

σd ≡ vertical planes which bisect the angles between adjacent vertical planes

⇒ Dnd point group

C734b Symmetry Operations and 46 Point groups

23 Special Cases

1.) Linear molecules: each molecule is its own axis of symmetry. order = ∞

C point group no σh: ⇒ ∞v

D point group σh exists: ⇒ ∞h

C734b Symmetry Operations and 47 Point groups

2.) Symmetries with > 1 high-order axis a) tetrahedron T point group Elements = {E, 8C3, 3C2, 6S4, 6σd} ⇒ d

C734b Symmetry Operations and 48 Point groups

24 b) Octahedron (Cubic Group)

2 Group elements: {E, 8C3, 6C2, 6C4, 3C2 (= C4 ), i, 6S4, 8S6, 3σh, 6σd}

⇒ Oh point group

C734b Symmetry Operations and 49 Point groups

c) Dodecahedron and icosahedron

2 3 Group elements: {E, 12C5, 12C5 , 20C3, 15C2, i, 12S10, 12S10 , 20S6, 15σ}

⇒ Ih point group (or Y point group)

icosahedron dodecahedron

C734b Symmetry Operations and 50 Point groups

25 There are 4 other point groups: T, Th, O, I which are not as important for molecules.

C734b Symmetry Operations and 51 Point groups

A systematic method for identifying point groups of any molecule

C734b Symmetry Operations and 52 Point groups

26