Structural Elucidation in Chemistry

1 CHEM2210

Structure Elucidation in Chemistry Prof Peter Gill

Part 1. Molecular Symmetry Symmetry Operations Point Groups Character Tables & Symmetry Species Orbital Symmetry

Part 2. Molecular Orbital Theory General Rules Diatomic Molecules Linear Polyatomic Molecules Non-Linear Polyatomic Molecules

Part 3. Molecular Vibrations and Electronic Transitions Reducible and Irreducible Representations Reduction Formula Application to Molecular Vibrations Application to Electronic Transitions

2

A molecule with symmetry has two or more orientations in space that are indistinguishable from one another.

Molecular symmetry is measured by the number and type of symmetry operations. Five different types of symmetry operations are important to us:

Symmetry Operations

3

1.  Identity (E). Trivial operation that does nothing.

2.  Proper Rotation (Cn). Rotation by 360º/n.

Symmetry Operations

4 Symmetry Operations

Two-fold C2 (180º), three-fold C3 (120º), four-fold C4 (90º), five-fold C5 (72º), six-fold C6 (60º), etc. proper rotations are possible.

The highest-fold rotation axis is known as the principal rotation axis of the molecule and is generally designated as the molecular z axis.

3.  Reflection (σ). A reflection in a mirror plane that maps one half of the molecule onto the other half.


There are three types of mirror planes:

Horizontal mirror plane (σh) is a mirror plane which is perpendicular to the principal rotation axis.


Vertical Mirror Plane (σv) is a mirror plane which contains the principal rotation axis.


Dihedral Mirror Plane (σd) is a mirror plane which contains the principal rotation axis and also bisects the angle between two C2 axes or two σv mirror planes.


4. Inversion ( i ). Turning a molecule “inside out” by moving each atom through a point (the centre of inversion) in a straight line to an identical atom on the other side at the same distance.


5. Improper Rotation (Sn). A proper rotation by 360o/n, followed by reflection through a plane perpendicular to the rotation axis.

S4 Improper Rotation Axis in CH4


Useful Websites for Learning about Symmetry

•  http://symmetry.otterbein.edu/

•  http://www.reciprocalnet.org/edumodules/symmetry/index.html

•  http://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_l3d.html

11 Point Groups

We can categorise molecules into point groups by finding the number and type of symmetry operations that they possess. Here are some examples:

C1 Only the identity operation E. Examples: CHFClBr and non-planar HOOCl

12 Point Groups

Ci Only a centre of inversion. Example: 1,2-dichloro-1,2-difluoro-ethane

13 Point Groups

Cs Only a mirror plane. Examples: CH2ClBr and HOCl

14 Point Groups

Cn Only an n-fold rotation axis. Examples: non-planar H2O2 (C2) and PPh3 (C3)

15 Point Groups

Cnv n-fold rotation axis and n vertical mirror planes. Examples: H2O (C2v) and NH3 (C3v)

16 Point Groups

Cnh n-fold rotation axis and horizontal mirror plane. Example: trans-C2H2Cl2 (C2h) & B(OCH3)3 (C3h)

17 Point Groups

Dn n-fold axis with n perpendicular 2-fold axes. Examples: Ph2 (D2) and Co(en)3

3+ (D3)

18 Point Groups

Dnd n-fold axis with n perpendicular two-fold axes and n dihedral planes Example: staggered C2H6 (D3d)

H

H H

H

H

C H

C C3

19 Point Groups

Dnh n-fold axis with n perpendicular two-fold axes and a horizontal mirror plane. Examples: C2H4 (D2h) and PF5 (D3h)

20 Point Groups

Td Four 3-fold axes and three 2-fold axes. Example: CH4

21 Point Groups

Oh Three perpendicular 4-fold axes, four 3-fold axes and a centre of inversion. Example: SF6

22

Dnh

Dnd

Cn? C2⊥Cn? σh?

σv ? σ ? σh? Cs

Yes Yes

No No No

Yes

Yes Yes Yes Cnh

i ? σv ? Dn

No No

Cnv

Cn C1

Ci Yes Yes

No

No No

Point Groups Point Groups Flow-Chart for Assigning Point Groups

Note: Start by testing for Td , Oh , C∞v and D∞h groups

23 Character Tables

Each point group has a character table that summarizes symmetry information for the group.

C2v E C2(z) σv(xz) σv'(yz)

A1 +1 +1 +1 +1 z x2, y2, z2 A2 +1 +1 -1 -1 Rz xy B1 +1 -1 +1 -1 x, Ry xz B2 +1 -1 -1 +1 y, Rx yz

classes of symmetry operations

symmetry species

basis functions a representation, Γ

a character, χ

24 Character Tables

We will use symmetry species to label molecular properties, e.g. orbitals, electronic states, vibrational modes, etc.

Each symmetry species is uniquely defined by a set of characters, χ, known as a representation, Γ.

The characters, χ, of a symmetry species tell us how an object (e.g. an orbital) is affected by symmetry operations in the point group.

25 Character Tables

If an object is symmetric (unchanged) with respect to a symmetry operation, its character is +1.

The dx²-y² orbital is unchanged by the C2(z) operation, so its character χ is +1.

26 Character Tables

If an object is antisymmetric (i.e. changes sign) with respect to a symmetry operation, its character is -1.

The px orbital changes sign under the C2(z) operation, so its character χ is -1.

27 Character Tables

Main symbols for the symmetry species mean:

A non-degenerate and symmetric with respect to the principal rotation axis.

B non-degenerate and antisymmetric with respect to the principal rotation axis.

E doubly-degenerate

T triply-degenerate

28 Character Tables



C3v E 2C3 3σv

A1 +1 +1 +1 z x2 +y2, z2 A2 +1 +1 -1 Rz E +2 -1 0 (x, y) (Rx, Ry) (x2 - y2, xy) (xz, yz)

29 Character Tables

Subscripts:

g symmetric with respect to inversion.

u antisymmetric with respect to inversion.

C2h E C2 i σh

Ag +1 +1 +1 +1 Rz x2, y2, z2, xy Bg +1 -1 +1 -1 Rx, Ry xz, yz Au +1 +1 -1 -1 z Bu +1 -1 -1 +1 x, y

30 Character Tables

Subscripts:

1 symmetric with respect to σv or (in the case of dihedral groups) a C2 axis perpendicular to the principal rotation axis.

2 antisymmetric with respect to σv or (in the case of dihedral groups) a C2 axis perpendicular to the principal rotation axis.

C3v E 2C3 3σv

A1 +1 +1 +1 z x2 +y2, z2 A2 +1 +1 -1 Rz E +2 -1 0 (x, y) (Rx, Ry) (x2 - y2, xy) (xz, yz)

31 Character Tables

Superscripts:

′ symmetric with respect to σh.

″ antisymmetric with respect to σh.

D3h E 2C3 3C2 σh 2S3 3σv

A1' +1 +1 +1 +1 +1 +1 x2 + y2, z2 A2' +1 +1 -1 +1 +1 -1 Rz E' +2 -1 0 +2 -1 0 (x, y) (x2 - y2, xy) A1" +1 +1 +1 -1 -1 -1 A2" +1 +1 -1 -1 -1 +1 z E" +2 -1 0 -2 +1 0 (Rx, Ry) (xz, yz)

32 Character Tables

Examples: A2′ a non-degenerate symmetry species, which is •  symmetric with respect to the principal axis •  antisymmetric with respect to σv •  symmetric with respect to σh

B1u a non-degenerate symmetry species, which is

•  antisymmetric with respect to principal axis

•  symmetric with respect to σv

•  antisymmetric with respect to inversion

33 Character Tables

Linear molecules with no inversion centre (e.g. CO)

C∞v E C2 2C4 … ∞σv Σ+ 1 +1 +1 … +1 z x2 + y2, z2 Σ− 1 +1 +1 … −1 Rz

Π 2 −2 0 … 0 (x,y) (Rx,Ry) (xz , yz) Δ 2 +2 −2 … 0 (x2 - y2, xy) : : : :

34 Character Tables

Linear molecules with an inversion centre (e.g. CO2)

D∞h E C2 2C4 … ∞σv i Σg

+ 1 +1 +1 … +1 +1 x2 + y2, z2 Σg− 1 +1 +1 … −1 +1 Rz

Πg 2 −2 0 … 0 +2 (Rx,Ry) (xz , yz) Δg 2 +2 −2 … 0 +2 (x2 - y2, xy) : : : : : Σu

+ 1 +1 +1 … +1 −1 z Σu− 1 +1 +1 … −1 −1

Πu 2 −2 0 … 0 −2 (x,y) Δu 2 +2 −2 … 0 −2 :

35 Character Tables

Do I have to memorize these character tables?

•  No. (Phew!)

•  All can be found in the “Symmetry Tables” file on Wattle

•  In an exam, any required character tables will be provided.

36 Orbital Symmetry

The rules for determining the character χ associated with a symmetry operation on an orbital are: 1.  If the operation moves the centre of the orbital,

then χ = 0.

37 Orbital Symmetry

2. If the operation swaps the orbital with another orbital on the same atom, then χ = 0.

38 Orbital Symmetry

3. If the operation leaves the orbital unchanged, then χ = +1.

39 Orbital Symmetry

4. If the operation changes the sign of the orbital, then χ is -1.

40 Orbital Symmetry

What are the characters of the oxygen 2p orbitals in the C2v water molecule?

41 Orbital Symmetry

χ = +1

χ = -1

χ = +1

χ = -1

42 Orbital Symmetry

χ = +1

χ = -1

χ = -1

χ = +1

43 Orbital Symmetry

χ = +1

χ = +1

χ = +1

χ = +1

44 Orbital Symmetry

C2v E C2 σv(xz) σv'(yz) Species

Γ(2px) +1 -1 +1 -1 B1

Γ(2py) +1 -1 -1 +1 B2

Γ(2pz) +1 +1 +1 +1 A1



45 Orbital Symmetry

An orbital belonging to a degenerate symmetry species (E or T) will not transform correctly in isolation. Degenerate orbitals must be considered collectively when generating the characters under a symmetry operation. Example: the 4p orbitals on bromine in BrF5 which has C4v symmetry:

C4v E 2C4 C2 2σv 2σd A1 +1 +1 +1 +1 +1 z x2+y2, z2 A2 +1 +1 +1 -1 -1 Rz

B1 +1 -1 +1 +1 -1 x2-y2

B2 +1 -1 +1 -1 +1 xy E +2 0 -2 0 0 (x, y) (Rx, Ry) (xz, yz)

46 Orbital Symmetry

47 Orbital Symmetry

48 Orbital Symmetry

C4v E 2C4 C2 2σv 2σd Species

Γ(pz) +1 +1 +1 +1 +1 A1

Γ(px) +1 0 -1 +1 0 -

Γ(py) +1 0 -1 -1 0 -

Γ(px+py) 2 0 -2 0 0 E

Individually, the px and py orbitals on the Br atom do not transform as a symmetry species of the C4v point group. However, together, they transform as the doubly degenerate E species.

In generating the above characters, the same σv and σd reflection planes are used for all three p orbitals.

49 Orbital Symmetry

The transformation of s, p and d orbitals on the central atom in a molecule can be determined directly from the point group character tables.

An s orbital always transforms as the totally symmetric representation (the first listed symmetry species with all χ = +1). The p and d orbitals transform the same as the following basis functions:

Orbital: px py pz dxy dxz dyz dx²-y² dz² Basis fn: x y z xy xz yz x2-y2 z2

In some point groups e.g. C2v, the dx²-y² orbital transforms as two separate functions x2 and y2.

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Structural Elucidation in Chemistry